A186266 Expansion of 2F1( 1/2, 3/2; 4; 16*x ).

1, 3, 18, 140, 1260, 12474, 132132, 1472328, 17065620, 204155380, 2506399896, 31443925968, 401783498480, 5215458874500, 68633685693000, 914099013896400, 12304253831789700, 167193096184907100, 2291164651422801000, 31637804708163654000, 439903041116118980400

Offset: 0

Author

Olivier Gérard, Feb 16 2011

Keywords

• nonn
• easy

Comments

Combinatorial interpretation welcome.

Could involve planar maps, lattice walks, and interpretations of Catalan numbers.

Links

• Indranil Ghosh, Table of n, a(n) for n = 0..800
• H. Franzen and T. Weist, The Value of the Kac Polynomial at One, arXiv preprint arXiv:1608.03419 [math.RT], 2016.

Crossrefs

Formula close to A000257, A000888, A172392.

Cf. A000108.

Programs

Mathematica

CoefficientList[
Series[HypergeometricPFQ[{1/2, 3/2}, {4}, 16*x], {x, 0, 20}], x]
Table[3 CatalanNumber[n] CatalanNumber[n+1] * (n+1) / (n+3), {n, 0, 20}] (* Indranil Ghosh, Mar 05 2017 *)

PARI

c(n) = binomial(2*n,n) / (n+1);
a(n) = 3 * c(n) * c(n+1) *(n+1) / (n+3); \\ Indranil Ghosh, Mar 05 2017

Python

import math
f=math.factorial
def C(n,r): return f(n) / f(r) / f(n-r)
def Catalan(n): return C(2*n, n) / (n+1)
def A186266(n): return 3 * Catalan(n) * Catalan(n+1) * (n+1) / (n+3) # Indranil Ghosh, Mar 05 2017

Formula

a(n) = 3*A000108(n)*A000108(n+1)*(n+1)/(n+3). - David Scambler, Aug 18 2012

D-finite with recurrence n*(n+3)*a(n) -4*(2*n-1)*(2*n+1)*a(n-1)=0. - R. J. Mathar, Jun 17 2016