A382874 Expansion of g.f. 2-hypergeom([3/2,7/2],[-1/2],4*x).

1, 42, 1890, 32340, 378378, 3567564, 29201172, 216164520, 1484052570, 9607866268, 59342703420, 352648983960, 2029131058500, 11360419371000, 62125264788840, 332868702695760, 1751865025825530, 9075126224864700, 46353422502086700, 233788539957892920

Offset: 0

Author

Karol A. Penson, Apr 07 2025

Keywords

• nonn

Crossrefs

Cf. A002423, A001700, A002457, A002421.

Programs

Maple

seq(coeff(series(2-hypergeom([3/2, 7/2], [-1/2], 4*x), x, k+1), x, k), k=0..19);

PARI

my(x='x+O('x^30)); Vec(2 - hypergeom([3/2,7/2],[-1/2],4*x)) \\ Michel Marcus, Apr 07 2025

Formula

a(0) = 1, a(n) = 8*4^n*(4*n^2 - 1)*Gamma(7/2 + n)/(15*sqrt(Pi)*n!), n>=1.

G.f.: 2 + (768*x^2 + 64*x - 1)/(1 - 4*x)^(11/2).

For n>=1, a(n) = (2*n-1) * (2*n+1)^2 * (2*n+3) * (2*n+5) * binomial(2*n,n)/15. - Vaclav Kotesovec, Apr 07 2025