A006633 - OEIS

A006633

Expansion of hypergeom([3/2, 7/4, 2, 9/4], [7/3, 8/3, 3], (256/27)*x).
(Formerly M4230)

1, 6, 39, 272, 1995, 15180, 118755, 949344, 7721604, 63698830, 531697881, 4482448656, 38111876530, 326439471960, 2814095259675, 24397023508416, 212579132600076, 1860620845932216, 16351267454243260, 144222309948974400, 1276307560533365955, 11329053395044653180

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OFFSET

0,2

COMMENTS

From generalized Catalan numbers.

REFERENCES

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

LINKS

Paolo Xausa, Table of n, a(n) for n = 0..1000

H. M. Finucan, Some decompositions of generalized Catalan numbers, 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.

Simon Plouffe, Approximations of generating functions and a few conjectures, Master's Thesis, arXiv:0911.4975 [math.NT], 2009.

FORMULA

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

a(n) = 2*binomial(4*n + 5, n) / (n+2). - Bruno Berselli, Jan 18 2014

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

MAPLE

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

ser := series(gf, x, 22): seq(coeff(ser, x, n), n = 0..21); # Peter Luschny, Feb 22 2024

MATHEMATICA

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

Array[A006633, 25, 0] (* Paolo Xausa, Feb 25 2024 *)

CROSSREFS