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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.

%I A287703 #14 Jul 28 2017 09:56:29
%S A287703 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,
%T A287703 0,-691,0,7,0,0,-2,0,691,0,-14,0,3617,0,1,0,-691,0,21,0,-25319,0,
%U A287703 43867,0,0,691,0,-10,0,75957,0,-438670,0,174611,0
%N 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.
%C A287703 For the rational triangle the reciprocals of the row sums are the Apéry numbers A005430.
%F A287703 A005430(n) = 1 / (Sum_{k=0..n-1} T(n,k)) for n>=1.
%e A287703 The rational triangle starts (with row sums at the end of the line):
%e A287703 1: [1/2], 1/2
%e A287703 2: [1/12, 0], 1/12
%e A287703 3: [0, 1/60, 0], 1/60
%e A287703 4: [-1/120, 0, 1/84, 0], 1/280
%e A287703 5: [0, -1/63, 0, 1/60, 0], 1/1260
%e A287703 6: [1/252, 0, -1/24, 0, 5/132, 0], 1/5544
%e A287703 7: [0, 1/40, 0, -5/33, 0, 691/5460, 0], 1/24024
%e A287703 8: [-1/240, 0, 7/44, 0, -691/936, 0, 7/12, 0], 1/102960
%e A287703 9: [0, -2/33, 0, 691/585, 0, -14/3, 0, 3617/1020, 0], 1/437580
%e A287703 The numerators of the triangle are:
%e A287703 1: [ 1]
%e A287703 2: [ 1,  0]
%e A287703 3: [ 0,  1,  0]
%e A287703 4: [-1,  0,  1,   0]
%e A287703 5: [ 0, -1,  0,   1,    0]
%e A287703 6: [ 1,  0, -1,   0,    5,   0]
%e A287703 7: [ 0,  1,  0,  -5,    0, 691, 0]
%e A287703 8: [-1,  0,  7,   0, -691,   0, 7,   0]
%e A287703 9: [ 0, -2,  0, 691,    0, -14, 0, 3617, 0]
%p A287703 T := (n,k) -> numer((-1)^n*binomial(n-1,k)*bernoulli(k+n)/(k+n)):
%p A287703 for n from 1 to 9 do seq(T(n,k), k=0..n-1) od;
%t A287703 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 *)
%o A287703 (PARI) T(n, k) = numerator((-1)^n*binomial(n-1,k)*bernfrac(k+n)/(k+n));
%o A287703 tabl(nn) = for (n=1, nn, for (k=0, n-1, print1(T(n, k), ", ")); print); \\ _Michel Marcus_, Jul 28 2017
%Y A287703 Cf. A005430 (Apéry), A287704 (denominators).
%K A287703 sign,tabl,frac
%O A287703 1,20
%A A287703 _Peter Luschny_, Jun 21 2017