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

Offset: 0

Author

Peter Bala, Oct 12 2011

Keywords

• nonn
• easy

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

• G. Schaeffer, A combinatorial interpretation of super-Catalan numbers of order two (2001).
• G. Thorlieffson, P. Bialis, B. Petersson, The weak-coupling limit of simplicial quantum gravity, Nuclear Physics B, Volume 550, Issues 1-2, 14 June 1999, Pages 465-491.

Crossrefs

Cf. A000139, A197271.

Programs

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 *)

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