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%I #17 Feb 08 2021 07:44:28
%S 1,2,9,56,420,3564,33033,327184,3413124,37119160,417733316,4837527072,
%T 57397642640,695394516600,8579210711625,107541060458400,
%U 1367139314643300,17599273282621800,229116465142280100,3013124257920348000,39991185556010816400
%N Expansion of 3F2( 2, 1/2, 3/2; 3, 4;16 x).
%C Combinatorial interpretation welcome.
%H Vincenzo Librandi, <a href="/A186262/b186262.txt">Table of n, a(n) for n = 0..200</a>
%F 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).
%F a(n) = C(2*n,n)*C(2*n+2,n)/C(n+3,3). - _Vaclav Kotesovec_, Oct 28 2012
%F 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
%t CoefficientList[Series[HypergeometricPFQ[{2, 1/2, 3/2}, {3, 4}, 16*x], {x, 0, 20}], x]
%t Table[Binomial[2*n,n]*Binomial[2*n+2,n]/Binomial[n+3,3],{n,0,20}] (* _Vaclav Kotesovec_, Oct 28 2012 *)
%Y Close to A138740.
%K nonn,easy
%O 0,2
%A _Olivier Gérard_, Feb 16 2011
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