Bernoulli's triangle

Bernoulli's triangle is an array of partial sums of the binomial coefficients.

[3] As the third column of Bernoulli's triangle (k = 2) is a triangular number plus one, it forms the lazy caterer's sequence for n cuts, where n ≥ 2.

[4] The fourth column (k = 3) is the three-dimensional analogue, known as the cake numbers, for n cuts, where n ≥ 3.

[5] The fifth column (k = 4) gives the maximum number of regions in the problem of dividing a circle into areas for n + 1 points, where n ≥ 4.

[6] In general, the (k + 1)th column gives the maximum number of regions in k-dimensional space formed by n − 1 (k − 1)-dimensional hyperplanes, for n ≥ k.[7] It also gives the number of compositions (ordered partitions) of n + 1 into k + 1 or fewer parts.

Derivation of Bernoulli's triangle (blue bold text) from Pascal's triangle (pink italics)
As the numbers of com­po­si­tions of n +1 into k +1 ordered partitions form Pascal's triangle , the numbers of compositions of n +1 into k +1 or fewer ordered partitions form Bernoulli's triangle
Sequences from the On-Line Encyclopedia of Integer Sequences in Bernoulli's triangle