A197272 a(n) = (6/((4*n+1)*(4*n+2)))*binomial(5*n,n).

3, 1, 3, 15, 95, 690, 5481, 46376, 411255, 3781635, 35791910, 346821930, 3427001253, 34425730640, 350732771160, 3617153918640, 37703805776935, 396716804816265, 4209161209968825, 44993046668984145, 484176486362971710, 5241928018191724860, 57064565236113814470

Offset

0,1

Comments

A combinatorial interpretation for this sequence in terms of a family of plane trees is given in [Schaeffer, Corollary 2 with k = 5].

A combinatorial interpretation for this sequence in terms of a family of four-dimensional stacked spheres is given in [Thorlieffson, Table 3 in Appendix B]. - Robert A. Russell, Mar 15 2012

Links

Table of n, a(n) for n=0..22.

Gilles Schaeffer, A combinatorial interpretation of super-Catalan numbers of order two, unpublished manuscript (2001).

G. Thorlieffson, P. Bialis, and B. Petersson, The weak-coupling limit of simplicial quantum gravity, Nuclear Physics B, Volume 550, Issues 1-2, 14 June 1999, Pages 465-491; alternative link.

Formula

a(n) = (6/((4*n+1)*(4*n+2)))*binomial(5*n,n).

D-finite with recurrence 8*n*(4*n+1)*(2*n+1)*(4*n-1)*a(n) - 5*(5*n-4)*(5*n-3)*(5*n-2)*(5*n-1)*a(n-1) = 0. - R. J. Mathar, Mar 29 2023

a(n) ~ 3 * 5^(5*n+1/2) / (2^(8*n+9/2) * n^(5/2) * sqrt(Pi)). - Amiram Eldar, Sep 17 2025

From Robert A. Russell, Oct 10 2025: (Start)

G.f.: 5*G(z) - 2*G(z)^2, where G(z)=1+z*G(z)^5 is the g.f. for A002294.

A233668(n-1) = n * a(n). (End)

Maple

A197272 := proc(n)
    6/((4*n+1)*(4*n+2))*binomial(5*n, n)
end proc:
seq(A197272(n), n=0..40) ; # R. J. Mathar, Mar 29 2023

Mathematica

Table[6/((4n+1)(4n+2)) Binomial[5n, n], {n, 0, 20}] (* Harvey P. Dale, Aug 08 2013 *)

Crossrefs

Cf. A000139, A001449, A002294, A197271, A233668.

Sequence in context: A143081 A179658 A112811 * A306773 A276639 A361840

Adjacent sequences: A197269 A197270 A197271 * A197273 A197274 A197275

Keyword

nonn,easy

Author

Peter Bala, Oct 12 2011

Status

approved