A292900 Triangle read by rows, a generalization of the Bernoulli numbers, the numerators for n>=0 and 0<=k<=n.

1, 0, 1, 0, -1, 1, 0, 1, -4, 0, 0, -1, 47, -10, -1, 0, 1, -221, 205, -209, 0, 0, -1, 953, -5495, 10789, -427, 1, 0, 1, -3953, 123445, -8646163, 177093, -22807, 0, 0, -1, 16097, -2534735, 22337747, -356249173, 3440131, -46212, -1

Offset: 0

Author

Peter Luschny, Oct 01 2017

Keywords

• sign
• tabl
• frac

Comments

The diagonal B(n, n) gives the Bernoulli numbers B_n = B_n(1). The formula is due to L. Kronecker and the generalization to Fukuhara, Kawazumi and Kuno.

Examples

			The triangle T(n, k) begins:
[0], 1
[1], 0,  1
[2], 0, -1,     1
[3], 0,  1,    -4,        0
[4], 0, -1,    47,      -10,       -1
[5], 0,  1,  -221,      205,     -209,          0
[6], 0, -1,   953,    -5495,    10789,       -427,       1
[7], 0,  1, -3953,   123445, -8646163,     177093,  -22807,      0
[8], 0, -1, 16097, -2534735, 22337747, -356249173, 3440131, -46212, -1
The rational triangle B(n, k) begins:
[0], 1
[1], 0,  1/2
[2], 0, -1/2,      1/6
[3], 0,  1/2,     -4/3,          0
[4], 0, -1/2,    47/12,      -10/3,         -1/30
[5], 0,  1/2,  -221/24,      205/9,       -209/20,          0
[6], 0, -1/2,   953/48,   -5495/54,      10789/80,    -427/10,       1/42
[7], 0,  1/2, -3953/96, 123445/324, -8646163/8640, 177093/200, -22807/105, 0
		

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, Über die Bernoullischen Zahlen, J. Reine Angew. Math. 94 (1883), 268-269.

Crossrefs

Cf. A292901 (denominators), B(n, n) = A164555(n)/A027642(n), A215083.

Programs

Maple

B := (n, k) -> `if`(n = 0, 1, add(((-1)^(j-n)/(j+1))*binomial(k+1, j+1)*add(i^n*(j-i+1)^(k-n), i=0..j), j=0..k)):
for n from 0 to 8 do seq(numer(B(n,k)), k=0..n) od;

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] // Numerator, {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 19 2018, from Maple *)

Formula

B(n, k) = Sum_{j=0..k}(((-1)^(j-n)/(j+1))*binomial(k+1, j+1)*Sum_{i=0..j}(i^n*(j-i+1)^(k-n))) if n >= 1 and B(0, 0) = 1.

B_n = B(n, n) = Sum_{j=0..n}((-1)^(n-j)/(j+1))*binomial(n+1,j+1)*(Sum_{i=0..j}i^n).

T(n, k) = numerator(B(n, k)).