A186264 - OEIS

%I #20 Feb 08 2021 07:47:21

%S 1,3,15,98,756,6534,61347,613470,6447012,70526404,797490876,

%T 9271926888,110380082000,1341117996300,16586474042475,208360804638150,

%U 2653858669601700,34220809160653500,446174168961282300,5875592302944678600,78078028942687784400

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

%C Combinatorial interpretation welcome.

%H Vincenzo Librandi, <a href="/A186264/b186264.txt">Table of n, a(n) for n = 0..200</a>

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

%F a(n) = 3/((n+3)*(n+2)^2)*(2*n+2)!^2/(n+1)!^4 = 3/(n+3)* Catalan(n+1)^2. - _Peter Bala_, Mar 28 2018

%F D-finite with recurrence (n+3)*(n+2)*a(n) -4*(2*n+1)^2*a(n-1)=0. - _R. J. Mathar_, Feb 08 2021

%p seq(3/((n+3)*(n+2)^2)*binomial(2*n+2,n+1)^2, n = 0..20); # _Peter Bala_, Mar 28 2018

%t CoefficientList[Series[HypergeometricPFQ[{1, 3/2, 3/2}, {3, 4}, 16*x], {x, 0, 20}],

%t   x]

%Y Cf. A186262, A000108, A001246.

%K nonn,easy

%O 0,2

%A _Olivier Gérard_, Feb 16 2011