A172392 - OEIS

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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

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OFFSET

0,2

LINKS

Table of n, a(n) for n=0..17.

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

EXAMPLE

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,...].

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

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 *)

PROG

(PARI) {a(n)=binomial(2*n, n)*binomial(2*n+2, n+1)/(n+2)}

CROSSREFS

Cf. A172391, A172393, A005568, A185248.

Sequence in context: A375172 A274665 A212073 * A127130 A052631 A368893

Adjacent sequences: A172389 A172390 A172391 * A172393 A172394 A172395

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Feb 05 2010

STATUS

approved