A348221 - OEIS

%I #13 Oct 08 2021 19:10:10

%S 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,

%T 3,1,90,1,3780,140,1,1,3,3,90,90,756,945,113400,1,1,3,3,90,45,756,756,

%U 16200,14175,1,1,3,1,90,10,3780,1,113400,1400,7484400

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

%C See A348220.

%D Paul Curtz, Intégration numérique des systèmes différentiels à conditions initiales. Note no. 12 du Centre de Calcul Scientifique de l'Armement, page 127, 1969.

%F Denominators of A(i,k) where:

%F A(i,k) = (1/k!)*Integral_(-1,1) Product(u+j, (j, -k+1 .. 0)) du for i=0.

%F A(i,k) = A(i-1, k-1) + A(i-1, k) for i>0.

%e Array begins:

%e 2, 0,    1/3,  -1/3,    29/90, -14/45,  1139/3780,   -41/140, ...

%e 2, 2,    1/3,     0,    -1/90,   1/90,   -37/3780,     8/945, ...

%e 2, 4,    7/3,   1/3,    -1/90,      0,      1/756,    -1/756, ...

%e 2, 6,   19/3,   8/3,    29/90,  -1/90,      1/756,         0, ...

%e 2, 8,   37/3,     9,   269/90,  14/45,   -37/3780,     1/756, ...

%e 2, 10,  61/3,  64/3,  1079/90,  33/10,  1139/3780,    -8/945, ...

%e 2, 12,  91/3, 125/3,  2999/90, 688/45, 13613/3780,    41/140, ...

%e 2, 14, 127/3,    72,  6749/90, 875/18,  14281/756,   736/189, ...

%e 2, 16, 169/3, 343/3, 13229/90,  618/5,  51031/756, 17225/756, ...

%e ...

%t 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;

%t Table[A[i - k, k] // Denominator, {i, 0, 10}, {k, 0, i}] // Flatten

%Y Cf. A002681, A002682, A348220 (numerators).

%K nonn,tabl,frac

%O 0,6

%A _Jean-François Alcover_ and _Paul Curtz_, Oct 08 2021