A292901 Triangle read by rows, a generalization of the Bernoulli numbers, the denominators for n>=0 and 0<=k<=n.
1, 1, 2, 1, 2, 6, 1, 2, 3, 1, 1, 2, 12, 3, 30, 1, 2, 24, 9, 20, 1, 1, 2, 48, 54, 80, 10, 42, 1, 2, 96, 324, 8640, 200, 105, 1, 1, 2, 192, 1944, 3840, 36000, 525, 35, 30, 1, 2, 384, 11664, 1244160, 720000, 756000, 3675, 168, 1
Offset: 0
Examples
Triangle starts: [0], 1 [1], 1, 2 [2], 1, 2, 6 [3], 1, 2, 3, 1 [4], 1, 2, 12, 3, 30 [5], 1, 2, 24, 9, 20, 1 [6], 1, 2, 48, 54, 80, 10, 42 [7], 1, 2, 96, 324, 8640, 200, 105, 1 [8], 1, 2, 192, 1944, 3840, 36000, 525, 35, 30 [9], 1, 2, 384, 11664, 1244160, 720000, 756000, 3675, 168, 1
Links
- S. Fukuhara, N. Kawazumi and Y. Kuno, Generalized Kronecker formula for Bernoulli numbers and self-intersections of curves on a surface, arXiv:1505.04840 [math.NT], 2015.
- L. Kronecker, Ueber die Bernoullischen Zahlen, J. Reine Angew. Math. 94 (1883), 268-269.
Programs
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Maple
# Function B(n,k) in A292900. for n from 0 to 9 do seq(denom(B(n, k)), k=0..n) od;
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Mathematica
B[0, 0] = 1; B[n_, k_] := Sum[(-1)^(j-n)/(j+1) Binomial[k+1, j+1] Sum[i^n (j-i+1)^(k-n), {i, 0, j}], {j, 0, k}]; Table[B[n, k] // Denominator, {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Feb 14 2019, from Maple *)
Comments