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A292901 Triangle read by rows, a generalization of the Bernoulli numbers, the denominators for n>=0 and 0<=k<=n.

%I A292901 #16 Aug 11 2025 05:23:24
%S A292901 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,
%T A292901 96,324,8640,200,105,1,1,2,192,1944,3840,36000,525,35,30,1,2,384,
%U A292901 11664,1244160,720000,756000,3675,168,1
%N A292901 Triangle read by rows, a generalization of the Bernoulli numbers, the denominators for n>=0 and 0<=k<=n.
%C A292901 See comments in A292900.
%H A292901 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 A292901 L. Kronecker, <a href="https://gdz.sub.uni-goettingen.de/id/PPN243919689_0094">Ueber die Bernoullischen Zahlen</a>, J. Reine Angew. Math. 94 (1883), 268-269.
%e A292901 Triangle starts:
%e A292901 [0], 1
%e A292901 [1], 1, 2
%e A292901 [2], 1, 2,   6
%e A292901 [3], 1, 2,   3,     1
%e A292901 [4], 1, 2,  12,     3,      30
%e A292901 [5], 1, 2,  24,     9,      20,      1
%e A292901 [6], 1, 2,  48,    54,      80,     10,     42
%e A292901 [7], 1, 2,  96,   324,    8640,    200,    105,    1
%e A292901 [8], 1, 2, 192,  1944,    3840,  36000,    525,   35,  30
%e A292901 [9], 1, 2, 384, 11664, 1244160, 720000, 756000, 3675, 168, 1
%p A292901 # Function B(n,k) in A292900.
%p A292901 for n from 0 to 9 do seq(denom(B(n, k)), k=0..n) od;
%t A292901 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 *)
%Y A292901 Cf. A292900 (numerators), T(n, n) = A027642(n).
%K A292901 nonn,tabl,frac
%O A292901 0,3
%A A292901 _Peter Luschny_, Oct 01 2017