login
A348220
Numerators of coefficients for numerical integration of certain differential systems (Array A(i,k) read by ascending antidiagonals).
1
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, 2, 12, 61, 9, 29, 0, -37, -41, 2, 14, 91, 64, 269, -1, 1, 8, 32377, 2, 16, 127, 125, 1079, 14, 1, -1, -119, -3956, 2, 18, 169, 72, 2999, 33, -37, 0, 127, 9, 2046263
OFFSET
0,1
COMMENTS
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.
Columns: A007395, A005843, A003215 (numerators).
REFERENCES
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.
FORMULA
Numerators of A(i,k) where:
A(i,k) = (1/k!)*Integral_(-1,1) Product(u+j, (j, -k+1 .. 0)) du for i=0.
A(i,k) = A(i-1, k-1) + A(i-1, k) for i>0.
EXAMPLE
Array begins:
2, 0, 1/3, -1/3, 29/90, -14/45, 1139/3780, -41/140, ...
2, 2, 1/3, 0, -1/90, 1/90, -37/3780, 8/945, ...
2, 4, 7/3, 1/3, -1/90, 0, 1/756, -1/756, ...
2, 6, 19/3, 8/3, 29/90, -1/90, 1/756, 0, ...
2, 8, 37/3, 9, 269/90, 14/45, -37/3780, 1/756, ...
2, 10, 61/3, 64/3, 1079/90, 33/10, 1139/3780, -8/945, ...
2, 12, 91/3, 125/3, 2999/90, 688/45, 13613/3780, 41/140, ...
2, 14, 127/3, 72, 6749/90, 875/18, 14281/756, 736/189, ...
2, 16, 169/3, 343/3, 13229/90, 618/5, 51031/756, 17225/756, ...
...
MATHEMATICA
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]];
A[_, _] = 0;
Table[A[i-k, k] // Numerator, {i, 0, 10}, {k, 0, i}] // Flatten
PROG
(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
CROSSREFS
Cf. A002681, A002682, A348221 (denominators).
Sequence in context: A349812 A029321 A029310 * A134131 A354186 A366265
KEYWORD
frac,sign,tabl
AUTHOR
STATUS
approved