OFFSET
0,2
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
EXAMPLE
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
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Feb 05 2010
STATUS
approved