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:
\( {\displaystyle \sum _{p=0}^{k}{n \choose p},} \)
i.e., the sum of the first k nth-order binomial coefficients.[1] The first rows of Bernoulli's triangle are:
\( {\displaystyle {\begin{array}{cc|cccccc}&k&0&1&2&3&4&5\\n&&\\\hline 0&&1\\1&&1&2\\2&&1&3&4\\3&&1&4&7&8\\4&&1&5&11&15&16\\5&&1&6&16&26&31&32\end{array}}} \)
Similarly to Pascal's triangle, each component of Bernoulli's triangle is the sum of two components of the previous row, except for the last number of each row, which is double the last number of the previous row. For example, if \( B_{{n,k}} \) denotes the component in row n and column k, then:
\( {\displaystyle {\begin{aligned}B_{n,k}=&B_{n-1,k}+B_{n-1,k-1}&{\mbox{ if }}&k<n\\B_{n,k}=&2B_{n-1,k-1}&{\mbox{ if }}&k=n\end{aligned}}} \)
As in Pascal's triangle and other similarly constructed triangles,[2] sums of components along diagonal paths in Bernoulli's triangle result in the Fibonacci numbers.[3]
References
On-Line Encyclopedia of Integer Sequences
Hoggatt, Jr, V. E., A new angle on Pascal's triangle, Fibonacci Quarterly 6(4) (1968) 221–234; Hoggatt, Jr, V. E., Convolution triangles for generalized Fibonacci numbers, Fibonacci Quarterly 8(2) (1970) 158–171
Neiter, D. & Proag, A., Links Between Sums Over Paths in Bernoulli's Triangles and the Fibonacci Numbers, Journal of Integer Sequences, 19 (2016) 16.8.3.
External links
The sequence of numbers formed by Bernoulli's triangle on the On-Line Encyclopedia of Integer Sequences: https://oeis.org/A008949.
Bernoulli's triangle
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
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