# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/ Search: id:a292900 Showing 1-1 of 1 %I A292900 #20 Jul 19 2018 08:23:29 %S A292900 1,0,1,0,-1,1,0,1,-4,0,0,-1,47,-10,-1,0,1,-221,205,-209,0,0,-1,953, %T A292900 -5495,10789,-427,1,0,1,-3953,123445,-8646163,177093,-22807,0,0,-1, %U A292900 16097,-2534735,22337747,-356249173,3440131,-46212,-1 %N A292900 Triangle read by rows, a generalization of the Bernoulli numbers, the numerators for n>=0 and 0<=k<=n. %C A292900 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. %H A292900 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. %H A292900 L. Kronecker, Über die Bernoullischen Zahlen, J. Reine Angew. Math. 94 (1883), 268-269. %F A292900 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. %F A292900 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). %F A292900 T(n, k) = numerator(B(n, k)). %e A292900 The triangle T(n, k) begins: %e A292900 [0], 1 %e A292900 [1], 0, 1 %e A292900 [2], 0, -1, 1 %e A292900 [3], 0, 1, -4, 0 %e A292900 [4], 0, -1, 47, -10, -1 %e A292900 [5], 0, 1, -221, 205, -209, 0 %e A292900 [6], 0, -1, 953, -5495, 10789, -427, 1 %e A292900 [7], 0, 1, -3953, 123445, -8646163, 177093, -22807, 0 %e A292900 [8], 0, -1, 16097, -2534735, 22337747, -356249173, 3440131, -46212, -1 %e A292900 The rational triangle B(n, k) begins: %e A292900 [0], 1 %e A292900 [1], 0, 1/2 %e A292900 [2], 0, -1/2, 1/6 %e A292900 [3], 0, 1/2, -4/3, 0 %e A292900 [4], 0, -1/2, 47/12, -10/3, -1/30 %e A292900 [5], 0, 1/2, -221/24, 205/9, -209/20, 0 %e A292900 [6], 0, -1/2, 953/48, -5495/54, 10789/80, -427/10, 1/42 %e A292900 [7], 0, 1/2, -3953/96, 123445/324, -8646163/8640, 177093/200, -22807/105, 0 %p A292900 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)): %p A292900 for n from 0 to 8 do seq(numer(B(n,k)), k=0..n) od; %t A292900 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}]; %t A292900 Table[B[n, k] // Numerator, {n, 0, 8}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Jul 19 2018, from Maple *) %Y A292900 Cf. A292901 (denominators), B(n, n) = A164555(n)/A027642(n), A215083. %K A292900 sign,tabl,frac %O A292900 0,9 %A A292900 _Peter Luschny_, Oct 01 2017 # Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE