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

%I A186266 #33 Feb 05 2025 15:07:55
%S A186266 1,3,18,140,1260,12474,132132,1472328,17065620,204155380,2506399896,
%T A186266 31443925968,401783498480,5215458874500,68633685693000,
%U A186266 914099013896400,12304253831789700,167193096184907100,2291164651422801000,31637804708163654000,439903041116118980400
%N A186266 Expansion of 2F1( 1/2, 3/2; 4; 16*x ).
%C A186266 Combinatorial interpretation welcome.
%C A186266 Could involve planar maps, lattice walks, and interpretations of Catalan numbers.
%H A186266 Indranil Ghosh, <a href="/A186266/b186266.txt">Table of n, a(n) for n = 0..800</a>
%H A186266 H. Franzen and T. Weist, <a href="https://arxiv.org/abs/1608.03419">The Value of the Kac Polynomial at One</a>, arXiv preprint arXiv:1608.03419 [math.RT], 2016.
%F A186266 a(n) = 3*A000108(n)*A000108(n+1)*(n+1)/(n+3). - _David Scambler_, Aug 18 2012
%F A186266 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
%t A186266 CoefficientList[
%t A186266 Series[HypergeometricPFQ[{1/2, 3/2}, {4}, 16*x], {x, 0, 20}], x]
%t A186266 Table[3 CatalanNumber[n] CatalanNumber[n+1] * (n+1) / (n+3), {n, 0, 20}] (* _Indranil Ghosh_, Mar 05 2017 *)
%o A186266 (PARI)
%o A186266 c(n) = binomial(2*n,n) / (n+1);
%o A186266 a(n) = 3 * c(n) * c(n+1) *(n+1) / (n+3); \\ _Indranil Ghosh_, Mar 05 2017
%o A186266 (Python)
%o A186266 import math
%o A186266 f=math.factorial
%o A186266 def C(n,r): return f(n) / f(r) / f(n-r)
%o A186266 def Catalan(n): return C(2*n, n) / (n+1)
%o A186266 def A186266(n): return 3 * Catalan(n) * Catalan(n+1) * (n+1) / (n+3) # _Indranil Ghosh_, Mar 05 2017
%Y A186266 Formula close to A000257, A000888, A172392.
%Y A186266 Cf. A000108.
%K A186266 nonn,easy
%O A186266 0,2
%A A186266 _Olivier Gérard_, Feb 16 2011