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

%I A292900 #22 Aug 11 2025 07:36:45
%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, <a href="https://arxiv.org/abs/1505.04840">Generalized Kronecker formula for Bernoulli numbers and self-intersections of curves on a surface</a>, arXiv:1505.04840 [math.NT], 2015.
%H A292900 L. Kronecker, <a href="https://gdz.sub.uni-goettingen.de/id/PPN243919689_0094">Über die Bernoullischen Zahlen</a>, 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