Wikipedia:Reference desk/Archives/Mathematics/2014 May 9

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May 9

sequence of polynomials

Is there a name for a sequence of polynomials, where the nth polynomial is the sum of (1-x)^n and x times the (n-1)th polynomial? So if the first polynomial is 1, the next few in the sequence are:

1 - x + x²

1 - 2x + 2x²

1 - 3x + 4x² - 2x³ + x⁴

1 - 4x + 7x² - 6x³ + 3x⁴

1 - 5x + 11x² - 13x³ + 9x⁴ - 3x⁵ + x⁶

1 - 6x + 16x² - 24x³ + 22x⁴ - 12x⁵ + 4x⁶

Arranged in a form similar to Pascal's triangle, each coefficient (ignoring signs) is the sum of two coefficients above it, other than the rightmost term. Mathew5000 (talk) 01:34, 9 May 2014 (UTC)

I'm not sure about a name, but here is the OEIS entry for this sequence. Sławomir Biały (talk) 11:48, 9 May 2014 (UTC)

Don't know, but rather than

${P}_{0}(x)=1$

${P}_{n}(x)=(1-x)^{n}+x{P}_{n-1}(x)$

you may find it more convenient to use

${P}_{0}(x)=1$

${P}_{1}(x)=1$

${P}_{n}(x)={P}_{n-1}(x)+x(x-1){P}_{n-2}(x)$

--catslash (talk) 13:12, 9 May 2014 (UTC)

Thank you very much, Sławomir Biały and Catslash! I had looked on OEIS and found some sequences that were relevant; A030441 gives values of alternate rows of the fourth column (i.e. coefficient of x³) with opposite sign, and A030442 gives values of alternate rows of the fifth column. Also, A003600 seems to give the fourth column, so that A030441 (after its first few terms) is a subset of A003600. But as far as I can see, none of those are cross-referenced with A220074 on OEIS so thanks very much for bringing that to my attention. Mathew5000 (talk) 19:42, 9 May 2014 (UTC)