# Talk:Pascal's triangle

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# Talk

## Why?

I was taught about pascal's triangle at an early age, and if I could give it a merit, it is good for a young mathematician as a learning tool in pattern recognition, but other than that, I cannot see the point. Euler's identity, whilst non-useful, is a point in the learning career of a mathematician where once seeing a proof of it and understanding it, one can sit back at the beauty of mathematics, but I had more of an "aaah, so that's how you do it" moment, when I discovered maclaurin's theorem and for the first time in my life figured out how to calculate a sine wave.

So, going back to my younger years.. why is this interesting? Looks like sudoku to me.. :-/ —Preceding unsigned comment added by 81.158.144.243 (talk) 00:30, 26 August 2009 (UTC)

This triangle is related with various common problems, like the sum of the n natural numbers from 1 to n and Metcalfe's_law. Also, there is a nice general formula for these sums: ${\displaystyle {\frac {(n+d)!}{n!d!}}}$, where n - number of elements and d - number of dimensions. It directly produces elements of our triangle
Thanks for the response! However, Metcalfe's law kinda post-dates it! What the hell was pascal up to? Incidentally, I'm currently writing a computer program involving fast Fourier transorms, and it took me a lot of reading to find out what Fourier was playing at by coming up with this stuff before the invention of the computer, but now I know, I find it highly unlikely that he could've made use of it in his analysis of steam engines! —Preceding unsigned comment added by 86.154.39.2 (talk) 20:23, 8 September 2010 (UTC)
The fast Fourier transform was invented long before computers okay - but by Gauss not Fourier. Dmcq (talk) 20:42, 8 September 2010 (UTC)

## Code

Hello, Iprogram with a much shorter, and, I hope, more comprehensible program. The goal here is to help people understand Pascal's triangle. I don't think the gyrations needed to get the numbers to line up were aiding that understanding. I also don't think the standard Java cruft (class name, imports, try/catch) was helping either. Anyway I'm sure that the floodgates are open now, and we're going to have 200 versions of this program in different languages... which is OK by me, as long as everybody is striving towards comprehension of Pascal's triangle. I claim the existing program is an improvement over the previous one. I don't claim it's the best possible. Happy editing, Wile E. Heresiarch 05:57, 8 Jun 2004 (UTC)

As I mentioned on Wikipedia: Wikicode/Specification, I really don't mind you reverting to the Python code; in fact, I probably shouldn't have converted it in the first place, since it's against my policy of leaving real-language code alone. Sorry about that. Derrick Coetzee 17:57, 10 Oct 2004 (UTC)

More about the code. I see the C code has been restored. I don't think this is an improvement. Why don't we just cut out the code altogther (in any language). Having an algorithm to print out some rows of the triangle doesn't have much to do with Pascal's triangle. I originally put in the Python because it replaced a Java program that was about 10 times as long; but in any event having a program is optional, so let's just cut it and avoid the language wars. Wile E. Heresiarch 06:47, 6 Mar 2005 (UTC)

So algorithms (though tied to a specific language) are not encyclopedia worthy? Cburnett 08:36, 6 Mar 2005 (UTC)
I agree with Wile that the code does not add any information on Pascal's triangle. The algorithm, based on Pascal's identity is already explained in English in the lead section, and it is straightforward to translate it in a specific programming language. -- Jitse Niesen 15:23, 6 Mar 2005 (UTC)
I've cut the section with the computer code. As I said in the edit summary, "source code doesn't shed any light on Pascal's triangle, and it's a language war magnet". For what it's worth, Wile E. Heresiarch 15:14, 7 Mar 2005 (UTC)

## Two Questions

1. Are the schemes for row numbering American and Candian??
2. Has anyone proposed a new scheme that has names with no ambiguity? 66.32.251.248 23:12, 17 Oct 2004 (UTC)
3. The History of Pascal's triangle as Pingala's work is highly dubious, the source cited is http://www.anaphoria.com/, i dont think thats a trustworthy source ?

1, 2. "Row n" contains the values of C(n,k) 3. It looks as though the commentary is being confused with the work itself so I've altered it. By the way, you are allowed to change the heading above! Xanthoxyl 19:53, 19 January 2007 (UTC)

i thought you only had 2 questions??? nvm joeldudesx20:14,19 February 2008(UTC) sry its 15 feb Joeldudesx (talk) 12:21, 15 February 2008 (UTC) joeldudes20:21 15 February 2008

## Question about a listed property

The page lists the following claim:

"the sum of the squares of the elements of the nth row equals the middle element of the 2nth."

But it appears to me that the claim should be "the sum of the squares of the elements of the nth row equals the middle element of the 2nth-1."

Am I wrong? This assumes that the row numbering starts at one (which is consistent with other parts of the text). In particular, only the odd numbered rows have a "middle element", and odd numbered rows must be of the form 2n-1, not 2n.

I believe you're right — this is an off-by-one error. Deco 08:37, 24 Mar 2005 (UTC)
Some further explanations prompted by the edits of Chad.nezar: The article assumes that the row with just one 1 is row number one. It then follows that row number n has the binomial coefficients
${\displaystyle {n-1 \choose 0},{n-1 \choose 1},{n-1 \choose 2},\ldots ,{n-1 \choose n-1}.}$
Hence, the text "the sum of the squares of the elements of the nth row equals the middle element of the (2n - 1)th" means
${\displaystyle \sum _{k=0}^{n-1}{n-1 \choose k}^{2}={2n-2 \choose n-1}.}$
Substituting ${\displaystyle m=n-1}$ yields
${\displaystyle \sum _{k=0}^{m}{m \choose k}^{2}={2m \choose m},}$
which is the formula given in the text. -- Jitse Niesen 10:57, 6 Apr 2005 (UTC)
Jitse, thanks for that example, which helps clarify some things for me. However, it also points out the other inconsistencies of this article. The crux of the problem is that the text uses the letter n to indicate rows as the nth row, where the row numbering starts at 1. However, clearly the math formulas and notation require that n start at 0. In the example you gave above, you have to define ${\displaystyle m=n-1}$ for just this purpose.
This truth is demonstrated by the example in this section, showing that squares of the terms of the 5th row (where row numbering starts from 1) add up to 70. However, the math formula demonstrating this sums over k which starts at zero. Thus, if n is equal to 5 (as the text would indicate), that sum has a total of 6 terms, not 5. Clearly it is written assuming that the nth row numbering starts with the 0th row.
In the first section, there is also some confusion in that the page states "for positive integers n and k where nk", however the examples require k to start at 0, and implicitly that n start at zero (otherwise, the case of ${\displaystyle {n \choose n}}$ is irrelevant). I therefore think the wording should be "for non-negative integers n and k where nk", with all the math formulas based on n starting at zero, and the text should be updated to not use the phrase "nth row", but something more appropriate. Either the text should be changed to indicate that we consider the first row to be the 0th row, or use another variable (like m) to indicate the row number and give it's relationship to n. Or just change nth row in the text, to ${\displaystyle (n+1)}$th row, where needed. Comments? Chad.netzer 22:07, 8 Apr 2005 (UTC)
I think your (Chad.netzer's) edits make the article clearer, so thanks for that. -- Jitse Niesen 11:51, 11 Apr 2005 (UTC)

## Khayyam-Pascal's triangle?

Any evidence that this is the "internationally recognized name" as stated at the beginning of the article? I grew up in Soviet Union, where it was the good old Pascal triangle, and the same in Europe and US. Oleg Alexandrov 21:37, 5 October 2005 (UTC)

Its always been Pascal's triangle in my part of the world. Paul August 23:19, 5 October 2005 (UTC)
Together with Charles at Wikipedia talk:WikiProject Mathematics#.5B.5BKhayyam-Pascal.27s_triangle.5D.5D, that makes three well-respected mathematicians, to which I add my own voice. Regardless of which name is more historically correct, we should use the name that is actually used. So I renamed it back to Pascal's triangle. I also put the history section in chronological order again, without removing the new contents. -- Jitse Niesen (talk) 23:26, 5 October 2005 (UTC)

In italian schools it is usually nameded Tartaglia's triangle.

The following is copied from my talk page:--Niels Ø 16:23, 9 October 2006 (UTC) Dear Noe I'm sending you this message to prevent Edit war. I added "or Kayyam-Pascal triangle " because by seeing this page at the first time I fell in doubt if it was that. I'm from Iran and we always call this triangle "Khayyam triangle" or "Khayyam-Pascal triangle". As you can see this in Persian(Farsi) Wikipedia[1] and tajik one [2]. And the other revalant languages(Such as Urdu, Pashto,Azeri,Kurdi,Uzbek,...) don't have this article yet. You also can watch Keeper Movie (The legend of Omar Khayyam)[3][4][5]. In addition you can have a look at its invention history. Khayyam had invented it sevral centuries before Pascal and applied it for the other applications. Pascal just showed new applications of this triangle in the absence of the knowing about khayyam invension. I think "or Kayyam-Pascal triangle " is strongly needed for removing disambiguation for hundred millions people. --81.31.160.22 07:29, 9 October 2006 (UTC)--Soroush ☺talk | ☼Contributions 07:32, 9 October 2006 (UTC)

My reply to User:Soroush_Mesry: I still disagree. This is an encyclopedia in English. The name of the triangle in English is Pascal's triangle. I suggest you create the relevant article (as a stub, for a start) in wikipedias in other relevant languages, and use interwiki links to connect things.
As for the history, I do not know whether Omar Khayyam invented the triangle. I don't think Pascal did. What we know for sure is that others invented it before any of them was born. So both names are misnomers, or at least give credit to the wrong people, but it's not our job to fix that - I mean, the name. What we can and should fix is giving credit to the right people when describing the history of the triangle, including the roles played by Khayyam, Pascal and others.
If names used in other languages must be listed at the top, the one used where you live is not the only one to be added.
PS. I am a bit of a Khayyam fan, but that does not affect the above.--Niels Ø 16:23, 9 October 2006 (UTC)
As can be gathered from my revert, I fully agree with Niels. -- Jitse Niesen (talk) 03:59, 10 October 2006 (UTC)
Agree too. Oleg Alexandrov (talk) 15:36, 10 October 2006 (UTC)

(See also the section #Recent changes to the lead further down on this talk page.)--Noe (talk) 13:26, 17 April 2008 (UTC)

I do not why you do not want to accept the reality and the fact.i accept that pascal's triangle is more popular than other names but we must respect the scientists and the one who was the first .
ya this page is an English page in wikipedia but it is not a good accuse for destroying the history and a great scientist.last time i added these terms to this page khayyam-pascal triangle and Tartaglia's triangle for respecting the scientist and the nations but again, you refuse to accept it and i got depressed because you as a mathematician do not respt the father's of mathematician and mathematics. —Preceding unsigned comment added by Roohollah1988 (talkcontribs) 21:41, 16 January 2010 (UTC)
It's still not our job to change the name, only to give credit to the right people, which we do in the lead (briefly) and in the history section (in more detail).-- (talk) 22:03, 16 January 2010 (UTC)
A very high proportion of theorems, perhaps most, in mathematics are named after the wrong person. However it is not Wikipedia's place to set the world right. What Wikipedia does is report on the world as as best it can from WP:reliable sources.. It is an encyclopaedia. It is not a place to argue over who first discovered something and then change the names given to things. If somebody does that elsewhere somehow the certainly Wikipedia can report it but not before. This is all part of WP:verifiability. Dmcq (talk) 22:14, 16 January 2010 (UTC)

It is well probable that Pascal was aware of Khayyam's work. i.e. is it a practice of plagiarism by Pascal? Anyway naming the triangle after Pascal is a misnomer and should be corrected. --Nevit (talk) 19:48, 22 April 2010 (UTC)

## Modifyed triangle "vandalism"

Recently, there was an edit to the modifyed triangle, by myself, that was referred to as vandalism and reverted as such. It is actually a mathematical correction. I invite anyone to check the math and find out for sure, and I will refrain from redoing the edit until someone can back me up, as I do not want to anger either the mod who made the edit nor the community as a whole. Aristotle2600 20:37, 22 October 2005 (UTC)

I reverted it. Thanks for bringing it up on the talk page, I should have done that myself. The change was first made several days ago here [6] by an anonymous user with no edit summary. To be perfectly honest, at least half if not more of anonymous edits are pure vandalism. So I checked the history of the page and the "84" value had been there for a long time (since the original version IIRC). On the balance of probabilities I therefore made the deciscion to revert the page on the likelihood that it was simple vandalism. Your change was the same as the anon's, only a day later, and I therefore decided to revert that also. Please please do use edit summaries if you make changes.
Understood. I'm still kinda new to Wikipedia, learning the finer details of etiquette and stuff; will try to remember edit summaries. Aristotle2600 20:22, 23 October 2005 (UTC)
If you just change a value with no justification in the edit summary it does look a bit suspicious. I am not mathematically inclined enough to calculate the value myself, so if you are happen to engage in dialog and are confident that it's correct value that's fine by me. I hope that explains things. chowells 21:36, 22 October 2005 (UTC)
I checked it and it indeed has to be 84. The rule in that triangle is that every value is twice the number to the upper left plus the number to the upper right, which makes 2 * 12 + 60 = 84. Thanks to Aristotle2600 for finding the mistake and to Chowells for doing RC patrol (though I wish you were a bit more into maths as a CompSci student ;) ). -- Jitse Niesen (talk) 00:31, 23 October 2005 (UTC)
Fine, thanks a lot for working it out. Yes I could have worked it out myself. I'm just lazy :) chowells 00:35, 23 October 2005 (UTC)

## Exponents

In the second equation of the 'Uses of Pascal's triangle' section the exponents are partially not in superscript. But this has been so from the very beginning. I'm no mathematician, but has this been overlooked all the time or am I missing something? DirkvdM 10:19, 11 November 2005 (UTC)

The equation look fine to me. The numbers immediately after the a should be subscripts (they indicate that a0 and a1 are different and unrelated variables, see the first paragraph of Index (mathematics)) and the numbers after the x and y should be superscripts, and that is how they appear to me. Perhaps a browser problem? -- Jitse Niesen (talk) 12:34, 11 November 2005 (UTC)
You're right, it's a browser problem. I used Konqueror, but in Mozilla it works ok. However, the font isn't too clear because they're in italic. Removing them also solves the problem with Konqueror. The italics don't seem necessary, so I've removed them. Correct me if I'm wrong. DirkvdM 11:32, 12 November 2005 (UTC)
It is customary to write variables in italics; look at any maths book and at Wikipedia:Manual of Style (mathematics). I never heard of Konqueror having these problems. I tried it out, using Konqueror 3.4.2 as included in Debian testing, but it renders the page correctly in my computer. Sorry, but it seems I can't help you. Groetjes, Jitse Niesen (talk) 18:12, 12 November 2005 (UTC)
I use Konqueror 3.2.1, so I suppose I need to update. It's quite weird. Number-exponents render just fine, as does (x+1)row number, but with (x+1)n+1 the 'n' isn't superscripted, but the '+1' is and with (x+1)n+1 they're both not superscripted, so it's purely a matter of italics. DirkvdM 08:12, 13 November 2005 (UTC)

## question

Where does (n+1) come from in the following line?

Note that the first row therefore corresponds to the binomial ${\displaystyle {0 \choose 0}}$, and can also be referred to as row ${\displaystyle (n+1)}$.
That's a good question. The answer is that I don't know. It has been there since 8 April. I removed the whole sentence. -- Jitse Niesen (talk) 12:46, 8 December 2005 (UTC)
In school, I haven't reached all of the complicated formulas. So I find it very confusing reading this article. There should really be some more explanation as to what the formulas mean. I'm just doing a patterning project and Pascal's triangle came up. I can't use any of the formulas, so there should be some morre explanation in words. I would have done this myself, but I don't know what any of this means. If someone could provide an explanation it would really help.

Yo, today 23:31, 5 February 2007 (UTC)

I just added the paragraph concerning the matrix-exponential. Since I'm not reading wikipedia daily, please mail a copy of any questions/critizism to me, so I could answer in time. Gottfried Helms --Gotti 12:03, 26 December 2005 (UTC)

What about moving that section to the Pascal matrix entry? The help page Wikipedia:Merging and moving pages does not seem to mention how to do this. Haseldon 18:22, 9 November 2006 (UTC)

## Properties of the triangle

Has anyone noticed this property?

2^0= 1 = 1

2^1= 1 + 1 = 2

2^2= 1 + 2 + 1 = 4

2^3= 1 + 3 + 3 + 1 = 8

2^4= 1 + 4 + 6 + 4 + 1 = 16

2^5= 1 + 5 + 10+10 + 5 + 1 = 32

-anon

That's in the article, at the bottom of the section on properties. Oleg Alexandrov (talk) 00:36, 29 December 2005 (UTC)

## patterns

i have noted and seen many patterns in pascal's triangle, but have not seen them documented in this article. i am at present not capable of cataloging or posting all or any of these, but i would appreciate it if someone could do that.

also, in searching the triangle, i noticed some particular parterns regarding perfect numbers, and wondered whether i was alone or not. i will list the patterns in a following post.

some help could be useful, thx

spartan60

There's probably an infinite number of patterns within Pascal's triangle. I don't think it's possible to document them all. It's mind-boggling how one simple mathematical structure has so much harmony. —Preceding unsigned comment added by 72.206.113.236 (talk) 02:03, 24 November 2008 (UTC)

## Mistake in triangle value?

I think the middle number in the last row should be 12870 (=16!/8!^2), not 12810. I've never edited anything here so I don't want to correct this myself.

You're right; I have fixed that.--Niels Ø 20:15, 5 November 2006 (UTC)
As I have noted at Image:Yanghui triangle.gif, there's also a mistake in that figure - an amusing little exercise to find it!--Niels Ø (noe) 20:18, 19 January 2007 (UTC)

## Relation to Sierpinski Triangle?

Could someone explain how this triangle is related to the Sierpinski Triangle here, and add a section on it to the article?--AeomMai 04:50, 2 January 2007 (UTC)

I think a "See also" reference would do, perhaps to Sierpinski Triangle#Properties.--Niels Ø (noe) 10:26, 2 January 2007 (UTC)

Isnt there an easier, more user friendly discription of the formulas on this page?

## Recent changes to the lead

I've made a post User_talk:Sangak/Archive3#Pascal.27s_triangle about the recent addition of Omar Khayyam to the lead. I guess that post really should have been here, but at least I've now linked to it.--Niels Ø (noe) 15:05, 25 February 2007 (UTC)

I did not notice the existence of a history section at the end of the long article. I've just noticed that some Indians and Italians also proposed similar ideas. I think the current version of the article is OK, although I see no point to have the history section at the end of the article. This is the first article I see with such a format. Sangak 15:22, 25 February 2007 (UTC)
I absolutely agree. It takes a bit more than just moving the history section, though, and I will not do it, at least not right now - go ahead if you are in the mood...--Niels Ø (noe) 15:31, 25 February 2007 (UTC)

if you want to see more boring math stuff give Finn i million cupcakes or youll explode — Preceding unsigned comment added by 205.244.38.4 (talk) 21:47, 12 December 2011 (UTC)

Hi,

I find it annoying to click at a link in wikipedia and get pointed to a commercial, closed-shop site like jstor.

At least the author could have introduced a separate section like "commercial links" or the like. Please remove that.

Gottfried Helms --Gotti 07:04, 26 February 2007 (UTC) —The preceding unsigned comment was added by Druseltal2005 (talkcontribs) 07:02, 26 February 2007 (UTC).

## Another property of the triangle or sequences

1/(1/999+1/999^2+1/999^3+1/999^4+1/999^5+1/999^6+1/999^7+1/999^8+.........infinity)=99800000....

                 1    9    45   165   495   1287  3003   6435   12870...
1    8   36   120   330   792   1716   3432   6435.....
1   7    28   84   210   462   924   1716   3003...
1   6   21   56   126   252   462   792   1287...
1   5   15   35   70   126   210   330   495.......
1    4   10  20   35   56    84    120   165..........
1    3   6   10   15   21   28    36    45..............
1  2    3   4    5    6    7     8    9...................
1   1   1   1   1    1    1    1     1......................

1   2    4   8   16   32   64   128   256


—Preceding unsigned comment added by Twentythreethousand (talkcontribs) 16:55, 24 May 2007

I'm not sure what the purpose of this post may be. The formula -
${\displaystyle {\frac {1}{{\frac {1}{999}}+{\frac {1}{999^{2}}}+{\frac {1}{999^{3}}}+\ldots }}={\frac {1}{\sum _{n=1}^{\infty }{\frac {1}{999^{n}}}}}=998}$
- is not obviously connected to Pascal's triangle. It follows from this:
The formula for a geometric series, ${\displaystyle a+ar+ar^{2}+\ldots =\sum _{n=1}^{\infty }a\times r^{n-1}={\frac {a}{1-r}}}$
implies ${\displaystyle \sum _{n=1}^{\infty }r^{n}={\frac {r}{1-r}}}$.
With ${\displaystyle r={\frac {1}{999}}}$, this gives the sum ${\displaystyle {\frac {\frac {1}{999}}{1-{\frac {1}{999}}}}={\frac {1}{999-1}}={\frac {1}{998}}}$.
The arrangement of the triangle shown next in the post seems to add nothing new.
The line sums being powers of two is covered in the article.--Niels Ø (noe) 09:58, 25 May 2007 (UTC)

Fibonacci number

1/999^1=0.001001001001001001001001001001001001001001001001001001001001001001001001001.....multiple of one
1/999^2=0.000001002003004005006007008009010011012013014015016017018019020021022023024.....
1/999^3=0.000000001003006010015021028036045055066078091105120136153171190210231253276.....
1/999^4=0.000000000001004010020035056084120165220286364455560680816970141331541773026.....
1/999^5=0.000000000000001005015035070126210330495716002366822383063880850992323865638.....
1/999^6=0.000000000000000001006021056126252462793289005007374196579643524375367691557.....
1/999^7=0.000000000000000000001007028084210462925719008013020394591170814338714081773.....
1/999^8=0.000000000000000000000001008036120330793719438446459479874465636450789503585.....
1/999^9=0.000000000000000000000000001009045165496290009447894353833708173810261050554.....
---1/998=0.001002004008016032064128256.....................................................

Pascal's triangle originated from one. 18:52,02 June 2007 twentythreethousand

square root of 98 twice:

3.14634628364578862062189264228281381561856023806624462402239289082033739605... close to Pi
Pi to the power of four equals 98-->two digits
98 to the power of two equals 9604-->four digits
9604 to the power of two equals 92236816-->eight digits
92236816 to the power of two equals 8507630225817856-->sixteen digits

The sequence keeps on going for every power of two.

01:02,03 June 2007 twentythreethousand

But ${\displaystyle \pi ^{4}=97.4}$, not 98.
And if we continue,
850763 02258178562 = 72 3797720592 4958347626 4088436736
(32 digits)
72 3797720592 4958347626 40884367362 = 5238 8314033489 2668972443 7407788362 7107149416 9665108966 4274333696
(64 digits)
5238 8314033489 2668972443 7407788362 7107149416 9665108966 42743336962 = 27445354 4727148846 0806819005 1661236507 3947642680 2443831829 2390078008 4434721144 6634073687 9050118395 5547012413 9598029526 4761020416
(128 digits)
27445354 4727148846 0806819005 1661236507 3947642680 2443831829 2390078008 4434721144 6634073687 9050118395 5547012413 9598029526 47610204162 = 75324 7482132970 9217345275 9423346606 6515260444 6475346020 2222838532 9110949199 7633094461 9885532021 3811976016 8733739899 9296698028 1048683891 3751212180 5911904487 0320727730 0983919820 4004316869 4824738055 1808978241 7363969167 8243422064 3303915197 5580337172 1568813056
(255 digits, not 256)
- anyway, what's the point? Are you suggesting a change to the article on Pascal's triangle here?--Niels Ø (noe) 09:15, 2 July 2007 (UTC)

twentythreethousand23:24 22 July 2007

Dear twentythreethousand. If you have changes to the article in mind, please either be bold and make them, or try to state precisely what you're suggesting.--Niels Ø (noe) 08:56, 23 July 2007 (UTC)

I maybe a long way from achieving what the numbers mean in the properties of Pascal's triangle, so saying that I wouldn't urge me to change the article on Pascal's triangle because there is not much to say other than the product of the polynomials equals to the inverse of eight.

twentythreethousand17:37 23 July 2007

## Relationship To Normal Distribution

Shouldn't we add something about how each row in Pascal's Triangle tends towards a normal distribution as we tend to the infinitieth row?

Martin Packer 09:40, 28 October 2007 (UTC)

## Vandalism

Just pointing out the vandalism at the end of section 3, immediately before the section on History. It's relatively minor, but should be removed.

Chris Cornwell 09:56, 11 November 2007 (UTC)

Can you please be more specific? I don't see any vandalism. -- Jitse Niesen (talk) 12:08, 12 November 2007 (UTC)
It was removed shortly after I posted, so it's gone. --Chris Cornwell 10:20, 12 November 2007 (UTC) —Preceding unsigned comment added by 24.11.210.232 (talk)

i think that would be a long time ago —Preceding unsigned comment added by Joeldudesx (talkcontribs) 12:10, 15 February 2008 (UTC)

### pascals triangle

who put that stuff there??? at pascals triangle that fibonacci numbers thingy or whatever and that supposed pic? of blaise pascal??? i hope someone will remove it...(Joeldudesx (talk) 12:19, 15 February 2008 (UTC)) joeldudesx/joeldudesx20:19 15 February 2008

### everybody listen up!!!

ok, can someone pls change the headline of square pyramidal numbers to pyramid numbers??? i had problem finding it! —Preceding unsigned comment added by Joeldudesx (talkcontribs) 12:23, 15 February 2008 (UTC)

### Replies (sort of)

Re "pascals triangle": What "stuff"? The Fibonacci numbers are really there; if you have a specific suggestion, question, criticism or whatever about that part of the article - well, please be more specific.

To sign your posts, place the cursor right after your post in the edit box, and click the signature button (the one with the mouse-over tip ""Your signature with timestamp") above the box.--Noe (talk) 16:36, 15 February 2008 (UTC)

## multiples of numbers

is it true that if i were to shade the multiples of each natural number in turn, would the pattern generated be of triangles every time? i tried out a few and they were all so, but i just wanted to make sure

## Another observation to pascal's triangle

• The sum of the entries in the nth row of binomial terms is the nth power of 2.The length of the binomial terms in the nth row(row 1 is 1, row 2 is 2,row 3 is 3 etc...)
• The sum of the entries in the nth row of trinomials is the nth power of 3.The length of the trinomial terms to every row to the power of nth follows a triangular number)
• The sum of the entries in the nth row of poynomials with four terms is the nth power of 4.The length of polynomials with four terms to the power of nth shows a Tetrahedral number.

etc... Twentythreethousand (talk) 19:54, 14 April 2008 (UTC)

Please write posts here as suggested changes, as discussion of specific parts of the article, or the like. What is it you want?--Noe (talk) 20:52, 14 April 2008 (UTC)

## Pascal's triangle or Tartaglia's triangle?

Here in Italy we always refer to it as Tartaglia's triangle and I'm adding a reference to it. I'm wondering if the whole article should be renamed to Tartaglia's triangle or Tartaglia-Pascal's triangle --82.48.35.113 (talk) 20:51, 11 June 2008 (UTC)

In English it is called Pascal's triangle, so we keep the most widespread usage. In Iran, it is called after a Persian mathematician, for example. See the history section in the article for more. Oleg Alexandrov (talk) 02:21, 12 June 2008 (UTC)

## odd numbers shared triangle has 2002 incorrectly shaded

the image for shaded odd numbers Pascal's_Triangle_divisible_by_2.svg has 2002 (row 15) incorrectly shaded --Firebladed (talk) 12:16, 24 June 2008 (UTC)

## Extensions

I have started an extensions section, but it needs cleaning up, and further expanding. Things I think it should contain are the current information on negative rows, with more detail with regard to the column formulae still applying when used with negative arguments. Also possible is to generalise the table to fractional rows, and I think this would also improve the article. Leahcim nai (talk) 03:00, 10 October 2008 (UTC)

## New use for the triangle

I am new to commenting on Wikipedia, so please forgive me.

I think I have found a new aspect of Pascal's Triangle. Please review this link, and perhaps add the information to Wikipedia:

http://www.mathhelpforum.com/math-help/probability-statistics/17147-combination-lock.html

My email is johnphantom@hotmail.com Johnphantom (talk) 21:13, 8 December 2008 (UTC)

## Question about binomial expansion proof

I don't understand how the proof that [the expansion of (x+1)n is described by Pascal's triangle] can be generalized to [(x+y)n is described by Pascal's triangle]. Just showing that it works in that specific case does not mean that it works in general...? --Ott0 (talk) 12:17, 19 January 2009 (UTC)

Edit: shortly afterwards I realized why, and have made a minor clarifying addition to the article for novices like myself. --Ott0 (talk) 00:04, 20 January 2009 (UTC)

We don't prove every theorem we mention in Wikipedia, sometimes just showing the rule and an example is enough--Tired time (talk) 19:42, 29 October 2009 (UTC)

## Errors in the 'divisible by X' triangles

In the bottom row of the 'divisible by X' triangles:

the number 380 (appears twice in the row) is incorrect, it should be 680. There might be more errors, I haven't checked all numbers.

The same error is in the figures in the section "Other patterns and properties", the fourth value should be 680, NOT 380. Pink18 (talk) 14:58, 14 October 2009 (UTC)

## more subtle patterns, repositioning of exp-picture

I moved the picture for the matrix-exponential back to the related paragraph. Also my *.png-version was reproduced as *.svg-file, which may benefituous for wikipedia. However, its display is always messed so I don't see the reason for that change. I removed that *svg-link and inserted the original *.png-link --Gotti 11:58, 29 July 2009 (UTC) —Preceding unsigned comment added by Druseltal2005 (talkcontribs)

## Fourier transform subsection

I edited this subsection to make it more encylopedic (dropped second person, etc.) and, I think, more understandable. In particular I tried to make it clear that the transform is a step function with certain values, not just a discrete sequence of numbers. However, I didn't actually carefully check the result itself (which is eminently plausible). In any case, it needs a reference. Anyone have one for this result? -- Spireguy (talk) 20:39, 25 September 2009 (UTC)

The first six rows of Pascal's triangle

It is said in the article that "The rows of Pascal's triangle are conventionally enumerated starting with row 0 ... On row 0, write only the number 1" So this means that in the lead image there are first five rows (not six), right? Normally I would just correct something like that, but it's a bit hard to believe that no one noticed such a mistake in the lead image of an important article for more than 5 years.--Tired time (talk) 18:14, 29 October 2009 (UTC)

The first row is "row 0", so the first six rows start with row 0 and end with row 5. The numbering seems to be done that way to match the notation of binomial coefficients. Xanthoxyl < 18:36, 29 October 2009 (UTC)
O yea, sorry--Tired time (talk) 19:42, 29 October 2009 (UTC)

## First 50 lines of Pascal's Triangle

1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1
1 7 21 35 35 21 7 1
1 8 28 56 70 56 28 8 1
1 9 36 84 126 126 84 36 9 1
1 10 45 120 210 252 210 120 45 10 1
1 11 55 165 330 462 462 330 165 55 11 1
1 12 66 220 495 792 924 792 495 220 66 12 1
1 13 78 286 715 1287 1716 1716 1287 715 286 78 13 1
1 14 91 364 1001 2002 3003 3432 3003 2002 1001 364 91 14 1
1 15 105 455 1365 3003 5005 6435 6435 5005 3003 1365 455 105 15 1
1 16 120 560 1820 4368 8008 11440 12870 11440 8008 4368 1820 560 120 16 1
1 17 136 680 2380 6188 12376 19448 24310 24310 19448 12376 6188 2380 680 136 17 1
1 18 153 816 3060 8568 18564 31824 43758 48620 43758 31824 18564 8568 3060 816 153 18 1
1 19 171 969 3876 11628 27132 50388 75582 92378 92378 75582 50388 27132 11628 3876 969 171 19 1
1 20 190 1140 4845 15504 38760 77520 125970 167960 184756 167960 125970 77520 38760 15504 4845 1140 190 20 1
1 21 210 1330 5985 20349 54264 116280 203490 293930 352716 352716 293930 203490 116280 54264 20349 5985 1330 210 21 1
1 22 231 1540 7315 26334 74613 170544 319770 497420 646646 705432 646646 497420 319770 170544 74613 26334 7315 1540 231 22 1
1 23 253 1771 8855 33649 100947 245157 490314 817190 1144066 1352078 1352078 1144066 817190 490314 245157 100947 33649 8855 1771 253 23 1
1 24 276 2024 10626 42504 134596 346104 735471 1307504 1961256 2496144 2704156 2496144 1961256 1307504 735471 346104 134596 42504 10626 2024 276 24 1
1 25 300 2300 12650 53130 177100 480700 1081575 2042975 3268760 4457400 5200300 5200300 4457400 3268760 2042975 1081575 480700 177100 53130 12650 2300 300 25 1
1 26 325 2600 14950 65780 230230 657800 1562275 3124550 5311735 7726160 9657700 10400600 9657700 7726160 5311735 3124550 1562275 657800 230230 65780 14950 2600 325 26 1
1 27 351 2925 17550 80730 296010 888030 2220075 4686825 8436285 13037895 17383860 20058300 20058300 17383860 13037895 8436285 4686825 2220075 888030 296010 80730 17550 2925 351 27 1
1 28 378 3276 20475 98280 376740 1184040 3108105 6906900 13123110 21474180 30421755 37442160 40116600 37442160 30421755 21474180 13123110 6906900 3108105 1184040 376740 98280 20475 3276 378 28 1
1 29 406 3654 23751 118755 475020 1560780 4292145 10015005 20030010 34597290 51895935 67863915 77558760 77558760 67863915 51895935 34597290 20030010 10015005 4292145 1560780 475020 118755 23751 3654 406 29 1
1 30 435 4060 27405 142506 593775 2035800 5852925 14307150 30045015 54627300 86493225 119759850 145422675 155117520 145422675 119759850 86493225 54627300 30045015 14307150 5852925 2035800 593775 142506 27405 4060 435 30 1
1 31 465 4495 31465 169911 736281 2629575 7888725 20160075 44352165 84672315 141120525 206253075 265182525 300540195 300540195 265182525 206253075 141120525 84672315 44352165 20160075 7888725 2629575 736281 169911 31465 4495 465 31 1
1 32 496 4960 35960 201376 906192 3365856 10518300 28048800 64512240 129024480 225792840 347373600 471435600 565722720 601080390 565722720 471435600 347373600 225792840 129024480 64512240 28048800 10518300 3365856 906192 201376 35960 4960 496 32 1
1 33 528 5456 40920 237336 1107568 4272048 13884156 38567100 92561040 193536720 354817320 573166440 818809200 1037158320 1166803110 1166803110 1037158320 818809200 573166440 354817320 193536720 92561040 38567100 13884156 4272048 1107568 237336 40920 5456 528 33 1
1 34 561 5984 46376 278256 1344904 5379616 18156204 52451256 131128140 286097760 548354040 927983760 1391975640 1855967520 2203961430 2333606220 2203961430 1855967520 1391975640 927983760 548354040 286097760 131128140 52451256 18156204 5379616 1344904 278256 46376 5984 561 34 1
1 35 595 6545 52360 324632 1623160 6724520 23535820 70607460 183579396 417225900 834451800 1476337800 2319959400 3247943160 4059928950 4537567650 4537567650 4059928950 3247943160 2319959400 1476337800 834451800 417225900 183579396 70607460 23535820 6724520 1623160 324632 52360 6545 595 35 1
1 36 630 7140 58905 376992 1947792 8347680 30260340 94143280 254186856 600805296 1251677700 2310789600 3796297200 5567902560 7307872110 8597496600 9075135300 8597496600 7307872110 5567902560 3796297200 2310789600 1251677700 600805296 254186856 94143280 30260340 8347680 1947792 376992 58905 7140 630 36 1
1 37 666 7770 66045 435897 2324784 10295472 38608020 124403620 348330136 854992152 1852482996 3562467300 6107086800 9364199760 12875774670 15905368710 17672631900 17672631900 15905368710 12875774670 9364199760 6107086800 3562467300 1852482996 854992152 348330136 124403620 38608020 10295472 2324784 435897 66045 7770 666 37 1
1 38 703 8436 73815 501942 2760681 12620256 48903492 163011640 472733756 1203322288 2707475148 5414950296 9669554100 15471286560 22239974430 28781143380 33578000610 35345263800 33578000610 28781143380 22239974430 15471286560 9669554100 5414950296 2707475148 1203322288 472733756 163011640 48903492 12620256 2760681 501942 73815 8436 703 38 1
1 39 741 9139 82251 575757 3262623 15380937 61523748 211915132 635745396 1676056044 3910797436 8122425444 15084504396 25140840660 37711260990 51021117810 62359143990 68923264410 68923264410 62359143990 51021117810 37711260990 25140840660 15084504396 8122425444 3910797436 1676056044 635745396 211915132 61523748 15380937 3262623 575757 82251 9139 741 39 1
1 40 780 9880 91390 658008 3838380 18643560 76904685 273438880 847660528 2311801440 5586853480 12033222880 23206929840 40225345056 62852101650 88732378800 113380261800 131282408400 137846528820 131282408400 113380261800 88732378800 62852101650 40225345056 23206929840 12033222880 5586853480 2311801440 847660528 273438880 76904685 18643560 3838380 658008 91390 9880 780 40 1
1 41 820 10660 101270 749398 4496388 22481940 95548245 350343565 1121099408 3159461968 7898654920 17620076360 35240152720 63432274896 103077446706 151584480450 202112640600 244662670200 269128937220 269128937220 244662670200 202112640600 151584480450 103077446706 63432274896 35240152720 17620076360 7898654920 3159461968 1121099408 350343565 95548245 22481940 4496388 749398 101270 10660 820 41 1
1 42 861 11480 111930 850668 5245786 26978328 118030185 445891810 1471442973 4280561376 11058116888 25518731280 52860229080 98672427616 166509721602 254661927156 353697121050 446775310800 513791607420 538257874440 513791607420 446775310800 353697121050 254661927156 166509721602 98672427616 52860229080 25518731280 11058116888 4280561376 1471442973 445891810 118030185 26978328 5245786 850668 111930 11480 861 42 1
1 43 903 12341 123410 962598 6096454 32224114 145008513 563921995 1917334783 5752004349 15338678264 36576848168 78378960360 151532656696 265182149218 421171648758 608359048206 800472431850 960566918220 1052049481860 1052049481860 960566918220 800472431850 608359048206 421171648758 265182149218 151532656696 78378960360 36576848168 15338678264 5752004349 1917334783 563921995 145008513 32224114 6096454 962598 123410 12341 903 43 1
1 44 946 13244 135751 1086008 7059052 38320568 177232627 708930508 2481256778 7669339132 21090682613 51915526432 114955808528 229911617056 416714805914 686353797976 1029530696960 1408831480060 1761039350070 2012616400080 2104098963720 2012616400080 1761039350070 1408831480060 1029530696960 686353797976 416714805914 229911617056 114955808528 51915526432 21090682613 7669339132 2481256778 708930508 177232627 38320568 7059052 1086008 135751 13244 946 44 1
1 45 990 14190 148995 1221759 8145060 45379620 215553195 886163135 3190187286 10150595910 28760021745 73006209045 166871334960 344867425584 646626422970 1103068603890 1715884494940 2438362177020 3169870830130 3773655750150 4116715363800 4116715363800 3773655750150 3169870830130 2438362177020 1715884494940 1103068603890 646626422970 344867425584 166871334960 73006209045 28760021745 10150595910 3190187286 886163135 215553195 45379620 8145060 1221759 148995 14190 990 45 1
1 46 1035 15180 163185 1370754 9366819 53524680 260932815 1101716330 4076350421 13340783196 38910617655 101766230790 239877544005 511738760544 991493848554 1749695026860 2818953098830 4154246671960 5608233007150 6943526580280 7890371113950 8233430727600 7890371113950 6943526580280 5608233007150 4154246671960 2818953098830 1749695026860 991493848554 511738760544 239877544005 101766230790 38910617655 13340783196 4076350421 1101716330 260932815 53524680 9366819 1370754 163185 15180 1035 46 1
1 47 1081 16215 178365 1533939 10737573 62891499 314457495 1362649145 5178066751 17417133617 52251400851 140676848445 341643774795 751616304549 1503232609100 2741188875410 4568648125690 6973199770790 9762479679110 12551759587400 14833897694200 16123801841600 16123801841600 14833897694200 12551759587400 9762479679110 6973199770790 4568648125690 2741188875410 1503232609100 751616304549 341643774795 140676848445 52251400851 17417133617 5178066751 1362649145 314457495 62891499 10737573 1533939 178365 16215 1081 47 1
1 48 1128 17296 194580 1712304 12271512 73629072 377348994 1677106640 6540715896 22595200368 69668534468 192928249296 482320623240 1093260079340 2254848913650 4244421484510 7309837001100 11541847896500 16735679449900 22314239266500 27385657281600 30957699535800 32247603683100 30957699535800 27385657281600 22314239266500 16735679449900 11541847896500 7309837001100 4244421484510 2254848913650 1093260079340 482320623240 192928249296 69668534468 22595200368 6540715896 1677106640 377348994 73629072 12271512 1712304 194580 17296 1128 48 1
1 49 1176 18424 211876 1906884 13983816 85900584 450978066 2054455634 8217822536 29135916264 92263734836 262596783764 675248872536 1575580702580 3348108992990 6499270398160 11554258485600 18851684897600 28277527346400 39049918716400 49699896548200 58343356817400 63205303218900 63205303218900 58343356817400 49699896548200 39049918716400 28277527346400 18851684897600 11554258485600 6499270398160 3348108992990 1575580702580 675248872536 262596783764 92263734836 29135916264 8217822536 2054455634 450978066 85900584 13983816 1906884 211876 18424 1176 49 1
1 50 1225 19600 230300 2118760 15890700 99884400 536878650 2505433700 10272278170 37353738800 121399651100 354860518600 937845656300 2250829575120 4923689695580 9847379391150 18053528883800 30405943383200 47129212244000 67327446062800 88749815264600 108043253366000 121548660036000 126410606438000 121548660036000 108043253366000 88749815264600 67327446062800 47129212244000 30405943383200 18053528883800 9847379391150 4923689695580 2250829575120 937845656300 354860518600 121399651100 37353738800 10272278170 2505433700 536878650 99884400 15890700 2118760 230300 19600 1225 50 1


I suppose for reference or something... —Preceding unsigned comment added by 87.194.102.100 (talk) 22:33, 9 November 2009 (UTC)

What's the point of all this? Are we to expect lines 51 to 100 next? Dmcq (talk) 22:39, 9 November 2009 (UTC)
51 rows are listed (row number 0 through 50) so the section title or content is incorrect anyway. Rick314 (talk) 21:29, 26 November 2009 (UTC)
Hm. Right you are. Okay 87.194.102.100 did you copythis from a book? If so the book is wrong about the 50 lines so it is harly a WP:Reliable source, if not it is WP:Original research. Just practicing some WP:wikilawyering here :) Dmcq (talk) 00:02, 27 November 2009 (UTC)
Btw, you have to include an empty line, otherwise the linebreak doesn't show up. I fixed it for you. Lanthanum-138 (talk) 13:32, 27 April 2011 (UTC)

## Al-Karaji worked in Baghdad, then capital of the Abbassid Caliphate

The text reports Al-Karaji being based in Persia (Iran), which I find misleading (and smacking of modern/silly Iranian--Arabic rivalry regarding history of sciences). Cerniagigante (talk) —Preceding undated comment added 08:47, 7 December 2009 (UTC).

## Rule 90

Dmcq has reverted the changes that 2 people here have pushed along. Yet this relationship between rule 90 and Pascal's triangle is well documented and references have been provided. It seems Dmcq applies his own, perhaps knowledgeable, judgment about it, but Wikipedia is not a primary source, we are only saying what it has been said and proven in the literature what there is to say about rule 90 w.r.t. Pascal's triangle. I propose to ask Dmcq not to act as if this article was of his own and if he has bibliographic objections or something to come here and ventilate them. Thanks. 189.146.51.246 (talk) —Preceding undated comment added 01:45, 14 December 2009 (UTC).

Please do not make WP:Personal attacks. You should WP:assume good faith if possible. The rule 90 addition was made at the same time as a change which said the pattern was the Sierpinski sieve rather than closely resembling it and growing more similar. I missed seeing the second change and for that I am sorry. As to the first change it is wrong and I will be changing it back again. Dmcq (talk) 08:29, 14 December 2009 (UTC)
Also rule 60 is what you get unless you put in some spacing. Rule 90 certainly looks nicer but it isn't what one gets directly, Dmcq (talk) 09:41, 14 December 2009 (UTC)
Thanks for your response Dmcq. Sorry for not having assumed good faith. If you have any suggestion before reversing or deleting please let me know. I will be happy to make the necessary changes to it if you are not willing to do so. Although I think you have already done it, not sure though. I'm mostly just citing the reference I'm giving. Thanks. 189.163.138.107 (talk) 17:42, 19 December 2009 (UTC)

## Sorting sections

The subsections of § Patterns and properties seemed to have no order other than presumably the sequence in which they have been written. I tried to sort them from simple (such as row properties) to complicated. I also created a section § Some other applications for the three sections: Number of elements of polytopes (which I renamed from "Geometric properties"), Fourier transform of sin(x)n+1/x, and Elementary cellular automaton. I started this section because I feel that the Fourier and Automaton texts don't contribute much to the knowledge of Pascal's triangle; they are somewhat like the famous "in popular culture" sections[7], formerly called "Trivia" section (see below). I then realized that the Polytope text would also fit in that section, since it is somewhat more complicated than the subsections of § Patterns and properties.

§ Elementary cellular automaton: The similarity to cellular automata is a trivial consequence of the fact that a cellular automaton is defined just the same as the construction of Pascal's triangle: Take two values, and add them. The only difference is that the automaton only looks at the least significant bit; that's already described in Addition#Addition in abstract algebra. Since elementary automata only give us one bit for each number, they provide no new information about Pascal's triangle.

Fourier transform of sin(x)n+1/x: This subsection discusses a straightforward consequence of the fact that Pascal's triangle can be seen as a repeated convolution, which is already discussed in § Relation to binomial distribution and convolutions. What makes it hard has nothing to do with any property of Pascal's triangle, but with the fact that the reader needs to be familiar with the Fourier transform, the sinc function, and convolution. But if you are already familiar with all that, then this discussion appears only as a needlessly specific limitation to one series of functions, sin(x)n+1/x. This is about as needlessly specific as adding "The fence around the back yard of the house in The Simpsons is wooden" to the wood article. There's nothing special about the boxcar function; we might as well add convolutions with a Triangular function, or a Gaussian.

For these reasons, I consider the latter two sections "Trivia" sections, and I would rather see them deleted from this article. — Sebastian 09:41, 12 February 2010 (UTC)

## Misuse of sources

A request for comments has been filed concerning the conduct of Jagged 85 (talk · contribs). That's an old and archived RfC. The point is still valid though. Jagged 85 is one of the main contributors to Wikipedia (over 67,000 edits, he's ranked 198 in the number of edits), and practically all of his edits have to do with Islamic science, technology and philosophy. This editor has persistently misused sources here over several years. This editor's contributions are always well provided with citations, but examination of these sources often reveals either a blatant misrepresentation of those sources or a selective interpretation, going beyond any reasonable interpretation of the authors' intent. I searched the page history, and found 18 edits by Jagged 85 in May 2007 and 3 more edits in March 2010. Tobby72 (talk) 21:22, 10 June 2010 (UTC)

Grrr.... You're quite right. The article says Al-Karaji provided the first proof of Pascals triangle, whatever that means, whereas the referenced source says he never provided any proofs just pictures. I'll just remove that bit since it was discovered earlier. Dmcq (talk) 22:07, 10 June 2010 (UTC)

## Number of elements of polytopes flaw

I noticed that in the section showing the relation between Pascal's Triangle and n-simplices, it states that the first row of ones represents a vertex in the next dimension as stated by this sentence "Thus, the meaning of the final number (1) in a row of Pascal's triangle becomes understood as representing the new vertex that is to be added to the simplex represented by that row to yield the next higher simplex represented by the next row." Since the 2nd row represnets 0-simplices (vertices), the 3rd row represents 1-simplices (line segments), the 4th row represents 2-simplicies (triangles) etc, tex in the next dimension. I would personally like to edit this, but would rather hear what other's have to say first. — Preceding unsigned comment added by Noctisi (talkcontribs) 06:51, 21 January 2011 (UTC)

I think the whole section should be reduced to a statement or two and a reference to Simplex and Hypercube. It is not cited and veers away from the topic. Dmcq (talk) 13:03, 21 January 2011 (UTC)

## Sum of the squares of the elements in a row.

I recently noticed a pattern in the rows of pascal's triangle. Please excuse my lack of TeX notation. Let us define a function f(n) (I know, generic, huh). f(n) equals the sum of the squares of the elements of row n. It seems to me that in my quick calculations by hand to about the 10th row, the limit as n approaches infinity, f(n) divided by f(n-1) approaches 4. I was hoping someone could help me prove this quickly and cleanly as to put it on the front page. —Preceding unsigned comment added by 68.112.252.38 (talk) 05:59, 1 February 2011 (UTC)

${\displaystyle \sum _{j=0}^{m}{\tbinom {m}{j}}^{2}={\tbinom {2m}{m}}}$ by equation 8 on this page so you get ${\displaystyle {\tbinom {2m+2}{m+1}}\div {\tbinom {2m}{m}}={\tfrac {(2m+2)(2m+1)}{(m+1)^{2}}}=4-{\tfrac {2}{m+1}}}$. Xanthoxyl < 16:05, 1 February 2011 (UTC)

## What is known about the highest number per row number sequence?

I'm talking about the sequence 1, 1, 2, 3, 6, 10, 20, 35, 70, 126, 252, with n being the highest value of the nth row. What is known about this sequence? Is there a formular to calculate the nth term? Does this sequence have an interesting sum of reciprocals or ratio between adjacent numbers like the other sequences in the triangle have? Robo37 (talk) 20:27, 9 September 2011 (UTC)

## Gray's Theory?

No relevant google hits on "Gray's theory". Is this actually a notable method? Seems maybe a bit OR to me. Staecker (talk) 19:14, 6 February 2013 (UTC)

Right, but rather than deleting the section I've removed the unsourced name ("Gray") and improved the presentation to use the same ${\displaystyle {n \choose k}}$ notation used elsewhere in the article. The material is not original as such, and I think the article benefits from having things of a fairly elementary nature in it too. But I wouldn't object strongly to the section being removed.-- (talk) 09:43, 20 March 2013 (UTC)

## Above the Triangle

What do we know about the extending the Triangle above the initial 1? There's a very interesting (and beautiful) line of reasoning here... Does anyone know of any formal mathematical investigations of this? — Preceding unsigned comment added by 98.229.181.98 (talk) 22:24, 9 February 2013‎

No, this (laterally symmetric) picture is not correct. Yes, there were investigations – see Beta function. Incnis Mrsi (talk) 07:29, 10 February 2013 (UTC)

Actually, the laterally symmetric picture would be correct if all the values above in the two upper triangles were halved. As it stands, using the regular recursion to work downwards, the top element in the normal triangle would be 2, not 1. Alternatively, either one or the other of the upper triangles should be all zeros - or you could take any weighted average of those two versions.

However, this is an encyclopedia, and this should of course only be included if it has proper weight and is properly sourced.-- (talk) 11:07, 10 February 2013 (UTC)

## Vulgarization

Would it be possible to include a short paragraph that goes into creating or making Pascal's triangle in simple terms ?

The page is more than ample when it comes information but it's uncertain that it would be of value to someone trying to assimilate the concept. It is introduced in lower secondary school so summation would not be known by students.

I realize that Wikipedia's aim is not to provide learning material but there is that aim of diffusing knowledge to the greater number. --JamesPoulson (talk) 17:30, 8 March 2013 (UTC)

I moved this new post to the bottom of the talk page as we usually do. However, it's actually related to the very first post (at his time), "Why?".-- (talk) 18:03, 8 March 2013 (UTC)

## Re: Verification Needed

The last sentence of the second paragraph in "History" currently states:

"In 1068, four columns of the first sixteen rows were given by the mathematician Bhattotpala, who realized the combinatorial significance.[4][verification needed]"

"A further commentator Bhattotpala (1068) has given an explicit example involving combinations of sixteen things (Fig.11)."

Bhattotpala is referred to by Edwards as a "commentator," not a mathematician. Whether he made any realization is not clear.

GiantSteps (talk) 05:10, 11 June 2013 (UTC)

Added this to the history sub-section of this article - "Varāhamihira was among the first mathematicians to discover a version of what is now known as the Pascal's triangle. He used it to calculate the binomial coefficients for evaluating combinatorics.". Sources are cited. isoham (talk) 20:21, 15 September 2013 (UTC)

## Failed to parse error

Am I the only one who sees a "failed to parse" error in the 'Binomial expansions' section? (starting after 'The two summations can be reorganized as follows:') Tropcho (talk) 17:46, 7 February 2014 (UTC)

Perhaps not, but I see the intended formula. I use Google Chrome - in what browser/version do you have the problem?-- (talk) 18:00, 7 February 2014 (UTC)

I see the problem both in Firefox 27 and Safari 6.0.5 on OS X Mountain Lion. Tropcho (talk) 19:15, 7 February 2014 (UTC)

Right, I see the same issue in Firefox on Windows 7. It reports that \begin is unknown. Looks to me like Chrome recognizes a larger subset of the TeX language used for formulae than Firefox. I don't know how to fix this, so I hope someone else do. Is there a way to draw the nattention of a wiki/tex expert to this issue?-- (talk) 10:39, 8 February 2014 (UTC)

## Usefulness of 1000th row image

1000th row of Pascal's triangle, arranged vertically, with grey-scale representations of decimal digits of the coefficients right-aligned. The left boundary of the dark segment-like shape corresponds roughly to the graph of the logarithm of the binomial coefficients, and illustrates that they form a log-concave sequence.

The 1000th row image (reproduced to the right) seems completely useless to me. I had to read the description several times before I really understood what it actually was (e.g., that the pixels represent decimal digits, or even the fact that it is "sideways" from the usual Pascal's triangle). There are no obvious patterns in the image, other than the symmetry along the middle. The only real thing that can be seen is the curved shape of the "length" of the numbers.

Since it is so confusing and doesn't really convey any useful information, I suggest removing it. I am going to remove it now, and have this discussion here if anyone wants to put it back.

asmeurer (talk | contribs) 21:23, 13 June 2014 (UTC)

I believe this might be the reference corresponding to this image (which I have removed from the main page). Bill Cherowitzo (talk) 21:59, 26 December 2014 (UTC)
Meeting: 1003, Atlanta, Georgia, SS 24A, AMS Special Session on Design Theory and Graph Theory, I 1003-06-607 Avery S. Zoch, Pascal’s Grey Scale

## Numbering of rows and columns

While I fully understand the awkwardness of it, the traditional manner of referring to the entries of the triangle is to say that C(n,k) appears in the nth row and kth position. The only way to make this work is to talk about the 0th row and 0th position in a row. This is what is done in most combinatorics texts and our job as editors is to report on what is in the literature and not try to make improvements on it. If you want to call the 0th row the "first row", then you will need to provide citations for that usage. I think a better approach would be to try to explain, in a more detailed manner, why mathematicians start the numbering with 0 in this case, so that the casual reader can see the reasoning behind this awkwardness. Bill Cherowitzo (talk) 20:26, 16 December 2014 (UTC)

Would it be acceptable to use "In the first row (n=0), ..." or should we use "In the zeroth row, ..."? I can see the latter being confusing but even more confusing is the present lack of consistency in the article. There are inconsistencies even in one sentence: "For example, the first number in the first row is 1 (the sum of 0 and 1), whereas the numbers 1 and 3 in the third row are added to produce the number 4 in the fourth row." (emphasis mine)
I'm somewhat partial to using "first row (n=0)," because it seems clearer to me. Is there a way to ensure that a standard is agreed upon and made obvious to readers and editors alike? (Sorry, I am somewhat new to editing Wikipedia)
Ezrysm (talk) 19:32, 24 December 2014 (UTC)

I think it is possible to be clearer about this. For instance, I'd use phrases like, "In the top (or zeroth) row, ..." and "... the initial number in the first row is 1 (in the zeroth position)." Using the parenthetic phrases should alert the reader that something unusual is going on without over emphasizing the issue. Bill Cherowitzo (talk) 00:32, 26 December 2014 (UTC)

## Notability

I have just removed the following new addition to the page:

### New properties : Inequalities

In 2014[1] were discovered new properties involving inequalities, which are:

1- In all the infinite center column of the triangle in the figure below, the product of two of its elements is greater than the product of two elements belonging to the same center column, located symmetrically between them. For example , in the figure below : 1 x 20 > 2 x 6 , or 2 x 20 > 6 x 6 , or yet 1 x 6 > 2 x 2. This applies to the entire central column.

                               1
1       1
1       2       1
1       3       3      1
1       4       6       4      1
1       5      10      10      5       1
1       6      15      20     15       6        1


2- Given two elements A and B in the infinite middle column, the product of these is greater than the product of two elements C and D belonging to the diagonal passing through A and B, which are located symmetrically in relation to A and B. For example, looking again the figure above: if a = 2 and B = 20 , then: 20 x 2 > 3 x 10 > 4 x 4 > 5 x 1 . If A = 1 and B = 20 , then: 20 x 1 > 1 x 10 > 1 x 4 > 1 x 1 .

There is a question of notability concerning this result. It is far too new to have received any reviews in the literature, so there are no secondary sources that attest to its notability. Wikipedia is not a place to announce new results. When this article has been vetted in the usual way, those secondary sources may be used as a basis for describing the result here. Even at that point, the above will need quite a bit of work to make it acceptable. Bill Cherowitzo (talk) 22:24, 7 April 2015 (UTC)

References

1. ^ Melo, L. A.; Santos, R. C.; Desigualdades no Triângulo de Pascal; Revista Eletrônica Paulista de Matemática. 2014. http://www2.fc.unesp.br/revistacqd/v3n1/v3n1_art7.pdf

## Wording, history

The history section begins:

The set of the numbers that form Pascal's triangle was known well before Pascal's time.

(Here, the second "the" has just been added by another editor.)

In a strict mathematical sense, the "set" of numbers in Pascal's triangle is all positive integers, which indeed were known... Am I nitpicking here, or should we rather have something like

The number patterns that make up Pascal's triangle were known well before Pascal's time.

Any views or better suggestions?-- (talk) 07:35, 5 October 2015 (UTC)

I like it and have made the change. --JBL (talk) 12:35, 5 October 2015 (UTC)

## Animated binary

Each frame represents a row in Pascal's triangle. Each column of pixels is a number in binary with the least significant bit at the bottom. Light pixels represent ones and the dark pixels are zeroes.

This gif image was added recently - frankly I don't think it adds anything significant. Other opinions? -- (talk) 09:54, 16 February 2016 (UTC)

: I don't see the point either, but nor do I see that it hurts the article. Since it takes no vertical space I'm for leaving it here. Gamall Wednesday Ida (talk) 07:07, 10 October 2016 (UTC)

## Why was Nilakantha Pi series discovery pulled

The Nilakantha Pi infinite series was discovered in Pascal's Triangle and has been sourced in a reputable journal, Mathematics Teacher, The National Council of Teachers of Mathematics. [1] Tonyfoster46 (talk) 20:41, 17 April 2016 (UTC)

Wikipedia does not publish original research results (see WP:NOR). Assuming that you are the author of the journal article, according to your edit you are the originator of this material which is first published in this journal. That makes the journal article a primary source for this material. In general, Wikipedia does not publish material based solely on primary sources. Reliable secondary sources (see WP:RSS) are needed to determine the value of the contribution and whether or not it should be included in Wikipedia. The author of the material has a clear bias and should not take part in this decision (see WP:COI). If, and when, your contribution has been reviewed and vetted by the mathematical community and its significance has been evaluated, we will be able to reconsider its inclusion in this article. Bill Cherowitzo (talk) 21:34, 17 April 2016 (UTC)

References

1. ^ Foster, Tony (2014). "Nilakantha Footprints In Pascal's Triangle". Mathematics Teacher. 108 (November): 247–248.

## Merge from Pascal's pyramid

I have suggested for the article Pascal's pyramid to be merged into this article as they are very closely related; but this article is much more improved an has less errors compared to the other article. 😃 Target360YT 😃 (talk · contribs) 06:01, 10 October 2016 (UTC)

Oppose.: I don't see how merging would suddenly improve the other article's contents. This one is already cluttered enough as it is, no need to add related subjects to it. The solution is to improve Pascal's pyramid, not to merge it here. Gamall Wednesday Ida (talk) 07:02, 10 October 2016 (UTC)

Oppose.: I totally agree with Gamall Wednesday Ida. Although mathematically related, the two topics are quite distinct and merging them would result in a bloated and confusing article.--Toploftical (talk) 23:24, 18 November 2016 (UTC)

Oppose: Both article have more than enough content on there own and though related they are nevertheless different subjects.--Kmhkmh (talk) 23:32, 16 January 2017 (UTC)

## Martin Gardner column on Pascal's triangle

I am reverting the revert by User:Gamall Wednesday Ida.

I added the sentence, "Martin Gardner wrote a popular account of Pascal's triangle in his December 1966 Mathematical Games column in Scientific American."

User Gamall reverted it saying, "How in the world is that important enough to deserve mention as part of the triangle's history?"

reason for revert of the revert

At the time that Gardner wrote the article, he had over a million readers and the column probably did more to make people aware of the triangle than anything previously written about it. In support of this statement we have

citations
books

Moreover, the column is included is included in several of Gardner's best selling books and continues to introduce the triangle to many readers to this day. All of this is certainly relevant to the history of the triangle in mathematics and human culture.

--Toploftical (talk) 18:20, 18 November 2016 (UTC)

Yes, those refs show that the column itself is generally notable as a popular culture phenomenon (and a good one, for once), but not that this specific instance of the column on this specific topic warrants mention. Is there a reliable ref arguing that there was a measurable effect on the use of the triangle after this column? Unless that's the case, I don't see how it rises above the level of trivia. The section goes straight from Pascal's posthumous publication, to de Moivre giving it its modern name slightly less than three centuries ago,... and therefrom straight to its appearing in a popular recreational maths column -- with no documented consequence whatsoever. One of those items does not belong with the others, especially in a section called "History".
Now, I have no specific animus against Gardner, who was, as far as I can tell, an admirable human being, and I'd be perfectly ok with the material if it came with references establishing the importance of the event for Pascal's triangle. As it is, it does not, so I'm not ok with it. Are you going to go over the list of Gardner's columns and make similar mentions in every article on every topic his column might have touched on? Or at least, would you do so if you had unlimited time and patience? That would be the logical consequence of your stated reasons, given that they are not specific to Pascal's triangle. — Gamall Wednesday Ida (t · c) 19:39, 18 November 2016 (UTC)
I agree entirely with GWI, excepting the last three sentences. Some things Martin Gardner wrote about are notable and their notability and significance can be traced to the fact that Gardner wrote about them, and for those things discussing Gardner's role in their history is a good idea. But Pascal's triangle is definitely not in this category: it is several centuries old, was well studied and recognized as important long before Gardner was born, and the fact that Gardner wrote about it has had no discernable effect on its modern history. --JBL (talk) 20:59, 18 November 2016 (UTC)
 Response to third opinion request: I agree that the sources cited support the claim that Gardner's column was, in general, influential, and I don't think there's any dispute about that. But what we need here is a source that shows that his column on the particular subject of Pascal's Triangle was influential by itself, and what, specifically, that influence was. What changed as a result of the column, and what reliable sources describe that change? Anaxial (talk) 14:03, 22 November 2016 (UTC)