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%I A006633 M4230 #40 Feb 25 2024 09:08:59
%S A006633 1,6,39,272,1995,15180,118755,949344,7721604,63698830,531697881,
%T A006633 4482448656,38111876530,326439471960,2814095259675,24397023508416,
%U A006633 212579132600076,1860620845932216,16351267454243260,144222309948974400,1276307560533365955,11329053395044653180
%N A006633 Expansion of hypergeom([3/2, 7/4, 2, 9/4], [7/3, 8/3, 3], (256/27)*x).
%C A006633 From generalized Catalan numbers.
%D A006633 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A006633 Paolo Xausa, <a href="/A006633/b006633.txt">Table of n, a(n) for n = 0..1000</a>
%H A006633 H. M. Finucan, <a href="https://doi.org/10.1007/BFb0061984">Some decompositions of generalized Catalan numbers</a>, pp. 275-293 of Combinatorial Mathematics IX. Proc. Ninth Australian Conference (Brisbane, August 1981). Ed. E. J. Billington, S. Oates-Williams and A. P. Street. Lecture Notes Math., 952. Springer-Verlag, 1982.
%H A006633 Simon Plouffe, <a href="http://arxiv.org/abs/0911.4975">Approximations of generating functions and a few conjectures</a>, Master's Thesis, arXiv:0911.4975 [math.NT], 2009.
%F A006633 O.g.f.: hypergeom_4F3([3/2, 7/4, 2, 9/4], [7/3, 8/3, 3], (256/27)*x). - _Simon Plouffe_, Master's Thesis, UQAM 1992
%F A006633 a(n) = 2*binomial(4*n + 5, n) / (n+2). - _Bruno Berselli_, Jan 18 2014
%F A006633 a(n) = (n+1) * A000260(n+1). - _F. Chapoton_, Feb 22 2024
%p A006633 gf := hypergeom([3/2, 7/4, 2, 9/4], [7/3, 8/3, 3], (256/27)*x):
%p A006633 ser := series(gf, x, 22): seq(coeff(ser, x, n), n = 0..21); # _Peter Luschny_, Feb 22 2024
%t A006633 A006633[n_] := 2*Binomial[4*n+5, n]/(n+2);
%t A006633 Array[A006633, 25, 0] (* _Paolo Xausa_, Feb 25 2024 *)
%Y A006633 Cf. A006631, A006632, A006634, A006635, A000027, A000260.
%K A006633 nonn
%O A006633 0,2
%A A006633 _Simon Plouffe_
%E A006633 New name by using a formula from the author by _Peter Luschny_, Feb 24 2024

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