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:

i.e., the sum of the first k nth-order binomial coefficients.[1] The first rows of Bernoulli's triangle are:

In the same way as Pascal's triangle, each component of Bernoulli's triangle is the sum of two components of the previous row, i.e., if denotes the component in row n and column k, then:

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

  1. ↑ On-Line Encyclopedia of Integer Sequences
  2. ↑ 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
  3. ↑ 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.
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