A006633 Expansion of hypergeom([3/2, 7/4, 2, 9/4], [7/3, 8/3, 3], (256/27)*x).

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

Offset: 0

Author

Simon Plouffe

Keywords

• nonn

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.

Crossrefs

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

Programs

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 *)

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

Extensions

New name by using a formula from the author by Peter Luschny, Feb 24 2024