A172392 a(n) = C(2n,n)*C(2n+2,n+1)/(n+2).
1, 4, 30, 280, 2940, 33264, 396396, 4907760, 62573940, 816621520, 10861066216, 146738321184, 2008917492400, 27815780664000, 388924218927000, 5484594083378400, 77926940934668100, 1114620641232714000
Offset: 0
Keywords
Examples
G.f.: A(x) = 1 + 4*x + 30*x^2 + 280*x^3 + 2940*x^4 + 33264*x^5 +... A(x) = 1 + 2*2*x + 5*6*x^2 + 14*20*x^3 + 42*70*x^4 + 132*252*x^5 +... where A(x)^2 = G(x*A(x)^2) and G(x) = A(x/G(x))^2 = g.f. of A172391: A172391=[1,8,12,0,28,0,264,0,3720,0,63840,0,1232432,0,25731216,0,...].
Programs
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Maple
A172392 := n -> 4^n*coeff(simplify(hypergeom([3/2, -2*n], [3], -x)),x,n): seq(A172392(n), n=0..17); # Peter Luschny, Feb 03 2015
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Mathematica
CoefficientList[ Series[HypergeometricPFQ[{1/2, 3/2}, {3}, 16 x], {x, 0, 20}], x] (* From Olivier Gérard, Feb 15 2011 *) Table[(Binomial[2n,n]Binomial[2n+2,n+1])/(n+2),{n,0,30}] (* Harvey P. Dale, Jul 16 2012 *) -
PARI
{a(n)=binomial(2*n,n)*binomial(2*n+2,n+1)/(n+2)}
Formula
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.
G.f. A(X) satisfies: A(x) = G(x*A(x)^2) and G(x) = A(x/G(x)^2) = g.f. of A172393.
a(n) = (n+1)*A005568(n) = A000108(n+1)*A000984(n), where A000108 is the Catalan numbers and A000984 is the central binomial coefficients.
G.f. : 2F1( (1/2, 3/2); (3))(16 x). - Olivier Gérard Feb 15 2011
a(n) = 4^n*[x^n]hypergeom([3/2, -2*n], [3], -x). - Peter Luschny, Feb 03 2015
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