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.