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
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.
Programs
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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.
Comments