Bernoulli's triangle

Bernoulli's triangle is an array of partial sums of the binomial coefficients. For any non-negative integer n and for any integer k included between 0 and n, the component in row n and column k is given by:

$\sum _{p=0}^{k}{n \choose p}$,

i.e., the sum of the first k nth-order binomial coefficients. The first rows of Bernoulli's triangle are:

1
1	2
1	3	4
1	4	7	8
1	5	11	15	16
1	6	16	26	31	32

In the same way as Pascal's triangle, components in a given row of Bernoulli's triangle are the sum of two components of the previous row, . I.e., if $B_{n,k}$ denotes the component in row n and column k, then:

$B_{n,k}=B_{n-1,k}+B_{n-1,k-1}$.

Hence, Bernoulli's triangle may be seen as a generalization of Pascal's triangle. Indeed, similarly to Pascal's triangle as well as other generalized Pascal's triangles, sums of components along diagonal paths in Bernoulli's triangle result in the Fibonacci numbers

External links

• The sequence of numbers formed by Bernoulli's triangle on the On-Line Encyclopedia of Integer Sequences: https://oeis.org/A008949.