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

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

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

Sequence in context: A264232 A068765 A349531 * A153392 A253077 A231482

Adjacent sequences: A006630 A006631 A006632 * A006634 A006635 A006636

Keyword

nonn

Author

Simon Plouffe

Extensions

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

Status

approved