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%I #21 Oct 11 2021 18:42:40
%S 2,2,0,2,2,1,2,4,1,-1,2,6,7,0,29,2,8,19,1,-1,-14,2,10,37,8,-1,1,1139,
%T 2,12,61,9,29,0,-37,-41,2,14,91,64,269,-1,1,8,32377,2,16,127,125,1079,
%U 14,1,-1,-119,-3956,2,18,169,72,2999,33,-37,0,127,9,2046263
%N Numerators of coefficients for numerical integration of certain differential systems (Array A(i,k) read by ascending antidiagonals).
%C It can be noticed that the sequence A002681/A002682 shows as these 4 subsequences: A(i, 2i+2), -A(i, 2i+3), A(i+1, 2i+2) and A(i+2, 2i+3), for i >= 0.
%C Columns: A007395, A005843, A003215 (numerators).
%D Paul Curtz, Intégration numérique des systèmes différentiels à conditions initiales. Note no. 12 du Centre de Calcul Scientifique de l'Armement, page 127, 1969, Arcueil. Later CELAR. Now DGA Maitrise de l'Information 35170 Bruz.
%F Numerators of A(i,k) where:
%F A(i,k) = (1/k!)*Integral_(-1,1) Product(u+j, (j, -k+1 .. 0)) du for i=0.
%F A(i,k) = A(i-1, k-1) + A(i-1, k) for i>0.
%e Array begins:
%e 2, 0, 1/3, -1/3, 29/90, -14/45, 1139/3780, -41/140, ...
%e 2, 2, 1/3, 0, -1/90, 1/90, -37/3780, 8/945, ...
%e 2, 4, 7/3, 1/3, -1/90, 0, 1/756, -1/756, ...
%e 2, 6, 19/3, 8/3, 29/90, -1/90, 1/756, 0, ...
%e 2, 8, 37/3, 9, 269/90, 14/45, -37/3780, 1/756, ...
%e 2, 10, 61/3, 64/3, 1079/90, 33/10, 1139/3780, -8/945, ...
%e 2, 12, 91/3, 125/3, 2999/90, 688/45, 13613/3780, 41/140, ...
%e 2, 14, 127/3, 72, 6749/90, 875/18, 14281/756, 736/189, ...
%e 2, 16, 169/3, 343/3, 13229/90, 618/5, 51031/756, 17225/756, ...
%e ...
%t A[i_ /; i >= 0, k_ /; k >= 0] := A[i, k] = If[i == 0, (1/k!) Integrate[ Product[u+j, {j, -k+1, 0}], {u, -1, 1}], A[i-1, k-1] + A[i-1, k]];
%t A[_, _] = 0;
%t Table[A[i-k, k] // Numerator, {i, 0, 10}, {k, 0, i}] // Flatten
%o (PARI) array(nn) = {my(m = matrix(nn, nn)); for (k=0, nn-1, m[1, k+1] = bestappr(intnum(x=-1, 1, prod(j=1-k, 0, x+j)))/k!; ); for (j=1, nn-1, for (k=0, nn-1, m[j+1, k+1] = if (k>0, m[j,k], 0) + m[j, k+1];);); apply(numerator, m);} \\ _Michel Marcus_, Oct 08 2021
%Y Cf. A002681, A002682, A348221 (denominators).
%Y Cf. A003215, A005843, A007395.
%K frac,sign,tabl
%O 0,1
%A _Jean-François Alcover_ and _Paul Curtz_, Oct 08 2021
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