A375259 Expansion of (1 - 3*x + 6*x^2 - 4*x^3)/(1 - 2*x - 3*x^2)^(9/2).

1, 6, 42, 224, 1134, 5292, 23562, 100584, 415701, 1671670, 6570564, 25325664, 95982068, 358442280, 1321336152, 4815108288, 17367199983, 62063418186, 219942717918, 773542367136, 2701767769470, 9376778431020, 32353614992790, 111032853586200, 379152389532735

Offset: 0

Author

Seiichi Manyama, Aug 08 2024

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..1000

Crossrefs

Column k=5 of A091869 (with a different offset).

Cf. A005717, A375260.

Programs

PARI

my(N=30, x='x+O('x^N)); Vec((1-3*x+6*x^2-4*x^3)/(1-2*x-3*x^2)^(9/2))

Formula

a(n) = (binomial(n+5,4)/5) * Sum_{k=0..floor(n/2)} binomial(n+1,n-2*k) * binomial(2*k+1,k).

a(n) = (binomial(n+5,4)/5) * A005717(n+1).

a(n) = ((n+5)/(n*(n+2))) * ((2*n+1)*a(n-1) + 3*(n+4)*a(n-2)).