A348221 Denominators of coefficients for numerical integration of certain differential systems (Array A(i,k) read by ascending antidiagonals).

1, 1, 1, 1, 1, 3, 1, 1, 3, 3, 1, 1, 3, 1, 90, 1, 1, 3, 3, 90, 45, 1, 1, 3, 3, 90, 90, 3780, 1, 1, 3, 1, 90, 1, 3780, 140, 1, 1, 3, 3, 90, 90, 756, 945, 113400, 1, 1, 3, 3, 90, 45, 756, 756, 16200, 14175, 1, 1, 3, 1, 90, 10, 3780, 1, 113400, 1400, 7484400

Offset

0,6

Comments

See A348220.

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.

Links

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

Formula

Denominators 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] // Denominator, {i, 0, 10}, {k, 0, i}] // Flatten

Crossrefs

Cf. A002681, A002682, A348220 (numerators).

Sequence in context: A099545 A300867 A269301 * A132429 A046540 A123191

Adjacent sequences: A348218 A348219 A348220 * A348222 A348223 A348224

Keyword

nonn,tabl,frac

Author

Jean-François Alcover and Paul Curtz, Oct 08 2021

Status

approved