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A292900
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Triangle read by rows, a generalization of the Bernoulli numbers, the numerators for n>=0 and 0<=k<=n.
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1
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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
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OFFSET
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0,9
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COMMENTS
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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.
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LINKS
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FORMULA
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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)).
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EXAMPLE
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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
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MAPLE
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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;
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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