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%I #19 Feb 08 2021 06:43:04
%S 1,4,30,280,2940,33264,396396,4907760,62573940,816621520,10861066216,
%T 146738321184,2008917492400,27815780664000,388924218927000,
%U 5484594083378400,77926940934668100,1114620641232714000
%N a(n) = C(2n,n)*C(2n+2,n+1)/(n+2).
%F G.f. A(X) satisfies: A(x)^2 = G(x*A(x)^2) and G(x) = A(x/G(x))^2 = g.f. of A172391.
%F G.f. A(X) satisfies: A(x) = G(x*A(x)^2) and G(x) = A(x/G(x)^2) = g.f. of A172393.
%F a(n) = (n+1)*A005568(n) = A000108(n+1)*A000984(n), where A000108 is the Catalan numbers and A000984 is the central binomial coefficients.
%F G.f. : 2F1( (1/2, 3/2); (3))(16 x). - _Olivier Gérard_ Feb 15 2011
%F a(n) = 4^n*[x^n]hypergeom([3/2, -2*n], [3], -x). - _Peter Luschny_, Feb 03 2015
%F D-finite with recurrence a(n) = a(n-1)*( 4*(4*n^2-1)/(n*(n+2)) ) for n>=1. - _Peter Luschny_, Feb 04 2015
%e G.f.: A(x) = 1 + 4*x + 30*x^2 + 280*x^3 + 2940*x^4 + 33264*x^5 +...
%e A(x) = 1 + 2*2*x + 5*6*x^2 + 14*20*x^3 + 42*70*x^4 + 132*252*x^5 +...
%e where A(x)^2 = G(x*A(x)^2) and G(x) = A(x/G(x))^2 = g.f. of A172391:
%e A172391=[1,8,12,0,28,0,264,0,3720,0,63840,0,1232432,0,25731216,0,...].
%p A172392 := n -> 4^n*coeff(simplify(hypergeom([3/2, -2*n], [3], -x)),x,n):
%p seq(A172392(n), n=0..17); # _Peter Luschny_, Feb 03 2015
%t CoefficientList[
%t Series[HypergeometricPFQ[{1/2, 3/2}, {3}, 16 x], {x, 0, 20}], x] (* From Olivier Gérard, Feb 15 2011 *)
%t Table[(Binomial[2n,n]Binomial[2n+2,n+1])/(n+2),{n,0,30}] (* _Harvey P. Dale_, Jul 16 2012 *)
%o (PARI) {a(n)=binomial(2*n,n)*binomial(2*n+2,n+1)/(n+2)}
%Y Cf. A172391, A172393, A005568, A185248.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Feb 05 2010
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