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 A292900 Triangle read by rows, a generalization of the Bernoulli numbers, the numerators for n>=0 and 0<=k<=n. 1
 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 (list; table; graph; refs; listen; history; text; internal format)
 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 Cf. A292901 (denominators), B(n, n) = A164555(n)/A027642(n), A215083. Sequence in context: A123163 A194794 A317448 * A177893 A058305 A193717 Adjacent sequences:  A292897 A292898 A292899 * A292901 A292902 A292903 KEYWORD sign,tabl,frac AUTHOR Peter Luschny, Oct 01 2017 STATUS approved

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Last modified August 13 19:30 EDT 2020. Contains 336451 sequences. (Running on oeis4.)