A287703 Triangle read by rows, numerators of T(n,k) = (-1)^n*binomial(n-1,k)*Bernoulli(n+k)/ (n+k) for n>=1, 0<=k<=n-1.
1, 1, 0, 0, 1, 0, -1, 0, 1, 0, 0, -1, 0, 1, 0, 1, 0, -1, 0, 5, 0, 0, 1, 0, -5, 0, 691, 0, -1, 0, 7, 0, -691, 0, 7, 0, 0, -2, 0, 691, 0, -14, 0, 3617, 0, 1, 0, -691, 0, 21, 0, -25319, 0, 43867, 0, 0, 691, 0, -10, 0, 75957, 0, -438670, 0, 174611, 0
Offset: 1
Examples
The rational triangle starts (with row sums at the end of the line): 1: [1/2], 1/2 2: [1/12, 0], 1/12 3: [0, 1/60, 0], 1/60 4: [-1/120, 0, 1/84, 0], 1/280 5: [0, -1/63, 0, 1/60, 0], 1/1260 6: [1/252, 0, -1/24, 0, 5/132, 0], 1/5544 7: [0, 1/40, 0, -5/33, 0, 691/5460, 0], 1/24024 8: [-1/240, 0, 7/44, 0, -691/936, 0, 7/12, 0], 1/102960 9: [0, -2/33, 0, 691/585, 0, -14/3, 0, 3617/1020, 0], 1/437580 The numerators of the triangle are: 1: [ 1] 2: [ 1, 0] 3: [ 0, 1, 0] 4: [-1, 0, 1, 0] 5: [ 0, -1, 0, 1, 0] 6: [ 1, 0, -1, 0, 5, 0] 7: [ 0, 1, 0, -5, 0, 691, 0] 8: [-1, 0, 7, 0, -691, 0, 7, 0] 9: [ 0, -2, 0, 691, 0, -14, 0, 3617, 0]
Programs
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Maple
T := (n,k) -> numer((-1)^n*binomial(n-1,k)*bernoulli(k+n)/(k+n)): for n from 1 to 9 do seq(T(n,k), k=0..n-1) od;
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Mathematica
T[n_, k_]:=Numerator[(-1)^n*Binomial[n - 1, k] BernoulliB[k + n]/(k + n)]; Table[T[n, k], {n, 11}, {k, 0, n - 1}]//Flatten (* Indranil Ghosh, Jul 27 2017 *) -
PARI
T(n, k) = numerator((-1)^n*binomial(n-1,k)*bernfrac(k+n)/(k+n)); tabl(nn) = for (n=1, nn, for (k=0, n-1, print1(T(n, k), ", ")); print); \\ Michel Marcus, Jul 28 2017
Formula
A005430(n) = 1 / (Sum_{k=0..n-1} T(n,k)) for n>=1.
Comments