OFFSET
0,9
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.
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.
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)).
EXAMPLE
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
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 *)
CROSSREFS
KEYWORD
AUTHOR
Peter Luschny, Oct 01 2017
STATUS
approved