%I M4230 #40 Feb 25 2024 09:08:59

%S 1,6,39,272,1995,15180,118755,949344,7721604,63698830,531697881,

%T 4482448656,38111876530,326439471960,2814095259675,24397023508416,

%U 212579132600076,1860620845932216,16351267454243260,144222309948974400,1276307560533365955,11329053395044653180

%N Expansion of hypergeom([3/2, 7/4, 2, 9/4], [7/3, 8/3, 3], (256/27)*x).

%C From generalized Catalan numbers.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Paolo Xausa, <a href="/A006633/b006633.txt">Table of n, a(n) for n = 0..1000</a>

%H 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 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 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 a(n) = 2*binomial(4*n + 5, n) / (n+2). - _Bruno Berselli_, Jan 18 2014

%F a(n) = (n+1) * A000260(n+1). - _F. Chapoton_, Feb 22 2024

%p gf := hypergeom([3/2, 7/4, 2, 9/4], [7/3, 8/3, 3], (256/27)*x):

%p ser := series(gf, x, 22): seq(coeff(ser, x, n), n = 0..21); # _Peter Luschny_, Feb 22 2024

%t A006633[n_] := 2*Binomial[4*n+5, n]/(n+2);

%t Array[A006633, 25, 0] (* _Paolo Xausa_, Feb 25 2024 *)

%Y Cf. A006631, A006632, A006634, A006635, A000027, A000260.

%K nonn

%O 0,2

%A _Simon Plouffe_

%E New name by using a formula from the author by _Peter Luschny_, Feb 24 2024