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

0,1

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.

Links

Table of n, a(n) for n=0..65.

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).

Cf. A003215, A005843, A007395.

Sequence in context: A349812 A029321 A029310 * A134131 A354186 A366265

Adjacent sequences: A348217 A348218 A348219 * A348221 A348222 A348223

Keyword

frac,sign,tabl

Author

Jean-François Alcover and Paul Curtz, Oct 08 2021

Status

approved