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:
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
i.e., the sum of the first knth-order binomial coefficients.[1] The first rows of Bernoulli's triangle are:
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 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]
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)-dimensionalhyperplanes, for n ≥ k.[7] It also gives the number of compositions (ordered partitions) of n + 1 into k + 1 or fewer parts.[8]
^Hoggatt, Jr, V. E., A new angle on Pascal's triangle, Fibonacci Quarterly6(4) (1968) 221–234; Hoggatt, Jr, V. E., Convolution triangles for generalized Fibonacci numbers, Fibonacci Quarterly8(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.
bernoulli, triangle, array, partial, sums, binomial, coefficients, negative, integer, integer, included, between, component, column, given, derivation, blue, bold, text, from, pascal, triangle, pink, italics, displaystyle, choose, numbers, tions, into, ordered. 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 Derivation of Bernoulli s triangle blue bold text from Pascal s triangle pink italics p 0 k n p displaystyle sum p 0 k n choose p As the numbers of com po si tions 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 i e the sum of the first k nth order binomial coefficients 1 The first rows of Bernoulli s triangle are k 0 1 2 3 4 5 n 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 displaystyle begin array cc cccccc amp k amp 0 amp 1 amp 2 amp 3 amp 4 amp 5 n amp amp hline 0 amp amp 1 1 amp amp 1 amp 2 2 amp amp 1 amp 3 amp 4 3 amp amp 1 amp 4 amp 7 amp 8 4 amp amp 1 amp 5 amp 11 amp 15 amp 16 5 amp amp 1 amp 6 amp 16 amp 26 amp 31 amp 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 displaystyle 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 if k lt n B n k 2 B n 1 k 1 if k n displaystyle begin aligned B n k amp B n 1 k B n 1 k 1 amp mbox if amp k lt n B n k amp 2B n 1 k 1 amp mbox if amp k n end aligned Sequences from the On Line Encyclopedia of Integer Sequences in Bernoulli s triangle 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 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 8 References Edit 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 amp Proag A Links Between Sums Over Paths in Bernoulli s Triangles and the Fibonacci Numbers Journal of Integer Sequences 19 2016 16 8 3 A000124 Oeis A000125 Oeis A000127 Oeis A006261 Oeis A008861 Oeis External links EditThe sequence of numbers formed by Bernoulli s triangle on the On Line Encyclopedia of Integer Sequences https oeis org A008949 Retrieved from https en wikipedia org w index php title Bernoulli 27s triangle amp oldid 1122734050, wikipedia, wiki, book, books, library,
