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 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, 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,20 COMMENTS For the rational triangle the reciprocals of the row sums are the Apéry numbers A005430. LINKS Table of n, a(n) for n=1..66. FORMULA A005430(n) = 1 / (Sum_{k=0..n-1} T(n,k)) for n>=1. EXAMPLE 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] 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; 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 *) PROG (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 CROSSREFS Cf. A005430 (Apéry), A287704 (denominators). Sequence in context: A186716 A331039 A171915 * A316480 A099224 A136598 Adjacent sequences: A287700 A287701 A287702 * A287704 A287705 A287706 KEYWORD sign,tabl,frac AUTHOR Peter Luschny, Jun 21 2017 STATUS approved

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Last modified September 16 23:59 EDT 2024. Contains 375984 sequences. (Running on oeis4.)