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

Author

Jean-François Alcover and Paul Curtz, Oct 08 2021

Keywords

• nonn
• tabl
• frac

Comments

See A348220.

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.

Crossrefs

Cf. A002681, A002682, A348220 (numerators).

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

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.