A186262 Expansion of 3F2( 2, 1/2, 3/2; 3, 4;16 x).

1, 2, 9, 56, 420, 3564, 33033, 327184, 3413124, 37119160, 417733316, 4837527072, 57397642640, 695394516600, 8579210711625, 107541060458400, 1367139314643300, 17599273282621800, 229116465142280100, 3013124257920348000, 39991185556010816400

Offset: 0

Author

Olivier Gérard, Feb 16 2011

Keywords

• nonn
• easy

Comments

Combinatorial interpretation welcome.

The sequence (n+3)*a(n) is the diagonal of the symmetric table b(m,n) = C(m+n,m)*C(m+n+2,m)*(m+3)/C(m+3,3). This table seems to have integer coefficients. - F. Chapoton, Jun 13 2025

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200

Crossrefs

Close to A138740.

Programs

Mathematica

CoefficientList[Series[HypergeometricPFQ[{2, 1/2, 3/2}, {3, 4}, 16*x], {x, 0, 20}], x]
Table[Binomial[2*n,n]*Binomial[2*n+2,n]/Binomial[n+3,3],{n,0,20}] (* Vaclav Kotesovec, Oct 28 2012 *)

Formula

G.f. is equivalent to (-1 + 2F1(-3/2,-1/2;2;16*x) - 6*x*2F1(-1/2,1/2;3;16*x) )/(4*x^2).

a(n) = C(2*n,n)*C(2*n+2,n)/C(n+3,3). - Vaclav Kotesovec, Oct 28 2012

D-finite with recurrence +n*(n+3)*(n+2)*a(n) -4*(2*n+1)*(2*n-1)*(n+1)*a(n-1)=0. - R. J. Mathar, Feb 08 2021