A006635 a(n) = 4*binomial(4*n+11, n)/(n+4). (Formerly M4860)

1, 12, 114, 1012, 8775, 75516, 649264, 5593068, 48336171, 419276660, 3650774820, 31907617560, 279871768995, 2463161027292, 21747225841440, 192575673551584, 1710009515037060, 15223466050169520, 135853465827080970, 1215067013768834100, 10890252031152078585

Offset

0,2

Comments

Former name: From generalized Catalan numbers. - G. C. Greubel, Sep 02 2025

References

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

Links

G. C. Greubel, 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, UQAM 1992; arXiv:0911.4975 [math.NT], 2009.

Formula

G.f.: 4F3([3,7/2,15/4,13/4], [5,14/3,13/3], 256*x/27). - Simon Plouffe, Master's thesis, UQAM 1992

G.f.: g^12 where g is the g.f. of A002293. - Sean A. Irvine, May 25 2017

a(n) = 4*binomial(4*n+11, n)/(n+4). - G. C. Greubel, Sep 02 2025

a(n) ~ 2^(8*n+49/2) / (3^(3*n+23/2) * n^(3/2) * sqrt(Pi)). - Amiram Eldar, Sep 23 2025

Mathematica

terms = 20; g[_] = 0; Do[g[x_] = 1 + x g[x]^4 + O[x]^terms, terms];
CoefficientList[g[x]^12, x] (* Jean-François Alcover, Oct 07 2018, after Sean A. Irvine *)
Table[4*Binomial[4*n+11, n]/(n+4), {n, 0, 40}] (* G. C. Greubel, Sep 02 2025 *)

Prog

(Magma)
A006635:= func< n | 4*Binomial(4*n+11, n)/(n+4) >;
[A006635(n): n in [0..40]]; // G. C. Greubel, Sep 02 2025
(SageMath)
def A006635(n): return 4*binomial(4*n+11, n)//(n+4)
print([A006635(n) for n in range(41)]) # G. C. Greubel, Sep 02 2025

Crossrefs

Cf. A002293, A196678, A006633, A233658, A233666, A006634, A233667.

Sequence in context: A378504 A331515 A154237 * A062386 A080602 A333299

Adjacent sequences: A006632 A006633 A006634 * A006636 A006637 A006638

Keyword

nonn,easy

Author

Simon Plouffe

Extensions

More terms from Sean A. Irvine, May 25 2017

New name by G. C. Greubel, Sep 02 2025

Status

approved