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

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

Author

Jean-François Alcover and Paul Curtz, Oct 08 2021

Keywords

• frac
• sign
• tabl

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).

Examples

			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, ...
...
		

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.

Crossrefs

Cf. A002681, A002682, A348221 (denominators).

Cf. A003215, A005843, A007395.

Programs

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

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

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.