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 A186266 Expansion of 2F1( 1/2, 3/2; 4; 16*x ). 1
 1, 3, 18, 140, 1260, 12474, 132132, 1472328, 17065620, 204155380, 2506399896, 31443925968, 401783498480, 5215458874500, 68633685693000, 914099013896400, 12304253831789700, 167193096184907100, 2291164651422801000, 31637804708163654000, 439903041116118980400 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Combinatorial interpretation welcome. Could involve planar maps, lattice walks, interpretations of catalan numbers. LINKS Indranil Ghosh, Table of n, a(n) for n = 0..800 H. Franzen, T. Weist, The Value of the Kac Polynomial at One, arXiv preprint arXiv:1608.03419 [math.RT], 2016. FORMULA a(n) = 3*A000108(n)*A000108(n+1)*(n+1)/(n+3). - David Scambler, Aug 18 2012 Conjecture: n*(n+3)*a(n) -4*(2*n-1)*(2*n+1)*a(n-1)=0. - R. J. Mathar, Jun 17 2016 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 *) PROG (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 CROSSREFS Formula close to A000257, A000888, A172392. Sequence in context: A183363 A216492 A127129 * A260506 A193237 A325996 Adjacent sequences:  A186263 A186264 A186265 * A186267 A186268 A186269 KEYWORD nonn,easy AUTHOR Olivier Gérard, Feb 16 2011 STATUS approved

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Last modified September 28 08:33 EDT 2020. Contains 337394 sequences. (Running on oeis4.)