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A348220 Numerators of coefficients for numerical integration of certain differential systems (Array A(i,k) read by ascending antidiagonals).

%I A348220 #21 Oct 11 2021 18:42:40
%S A348220 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 A348220 2,12,61,9,29,0,-37,-41,2,14,91,64,269,-1,1,8,32377,2,16,127,125,1079,
%U A348220 14,1,-1,-119,-3956,2,18,169,72,2999,33,-37,0,127,9,2046263
%N A348220 Numerators of coefficients for numerical integration of certain differential systems (Array A(i,k) read by ascending antidiagonals).
%C A348220 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 A348220 Columns: A007395, A005843, A003215 (numerators).
%D A348220 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 A348220 Numerators of A(i,k) where:
%F A348220 A(i,k) = (1/k!)*Integral_(-1,1) Product(u+j, (j, -k+1 .. 0)) du for i=0.
%F A348220 A(i,k) = A(i-1, k-1) + A(i-1, k) for i>0.
%e A348220 Array begins:
%e A348220 2, 0,    1/3,  -1/3,    29/90, -14/45,  1139/3780,   -41/140, ...
%e A348220 2, 2,    1/3,     0,    -1/90,   1/90,   -37/3780,     8/945, ...
%e A348220 2, 4,    7/3,   1/3,    -1/90,      0,      1/756,    -1/756, ...
%e A348220 2, 6,   19/3,   8/3,    29/90,  -1/90,      1/756,         0, ...
%e A348220 2, 8,   37/3,     9,   269/90,  14/45,   -37/3780,     1/756, ...
%e A348220 2, 10,  61/3,  64/3,  1079/90,  33/10,  1139/3780,    -8/945, ...
%e A348220 2, 12,  91/3, 125/3,  2999/90, 688/45, 13613/3780,    41/140, ...
%e A348220 2, 14, 127/3,    72,  6749/90, 875/18,  14281/756,   736/189, ...
%e A348220 2, 16, 169/3, 343/3, 13229/90,  618/5,  51031/756, 17225/756, ...
%e A348220 ...
%t A348220 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 A348220 A[_, _] = 0;
%t A348220 Table[A[i-k, k] // Numerator, {i, 0, 10}, {k, 0, i}] // Flatten
%o A348220 (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 A348220 Cf. A002681, A002682, A348221 (denominators).
%Y A348220 Cf. A003215, A005843, A007395.
%K A348220 frac,sign,tabl
%O A348220 0,1
%A A348220 _Jean-François Alcover_ and _Paul Curtz_, Oct 08 2021