A197271 a(n) = (10 / ((3*n+1)*(3*n+2))) * binomial(4*n, n).

5, 2, 5, 20, 100, 570, 3542, 23400, 161820, 1159400, 8544965, 64448228, 495508780, 3872033900, 30680401500, 246041115600, 1993987498284, 16310419381080, 134519771966180, 1117653277802000, 9347742311507600, 78652006531467930, 665393840873409150, 5657273782416664200, 48318619683648190500

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 = 4].

For n>=1, the number of rooted strict triangulations of a square with n-1 internal vertices, where a triangulation is "strict" if no two distinct edges have the same pair of ends. See equation (1) in [Tutte 1980] (who references [Brown 1964]) for the number of rooted strict near-triangulations of type (n,m), with m=1. - Noam Zeilberger, Jan 04 2023

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..1031
• William G. Brown, Enumeration of Triangulations of the Disk, Proc. Lond. Math. Soc. s3-14 (1964) 746-768.
• W. G. Brown, Enumeration of Triangulations of the Disk, Proc. Lond. Math. Soc. s3-14 (1964) 746-768. [Annotated scanned copy]
• K. A. Penson, K. Górska, A. Horzela, and G. H. E. Duchamp, Hausdorff moment problem for combinatorial numbers of Brown and Tutte: exact solution, arXiv:2209.06574 [math.CO], 2022.
• G. Schaeffer, A combinatorial interpretation of super-Catalan numbers of order two, (2001).
• William T. Tutte, On the enumeration of convex polyhedra, J. Combin. Theory Ser. B 28 (1980), 105-126.

Crossrefs

Column m=1 of A146305.

Cf. A000139, A000260, A197272.

Programs

Mathematica

Table[10/((3n+1)(3n+2)) Binomial[4n,n],{n,0,30}] (* Harvey P. Dale, Jan 27 2015 *)

Formula

a(n) = 10/((3*n+1)*(3*n+2))*binomial(4*n,n).

a(n) = A000260(n) * 5*(n+1)/(4*n+1). - Noam Zeilberger, May 20 2019

a(n) ~ c*(256/27)^n / n^(5/2), where c = (10/9)*sqrt(2/(3*Pi)) = 0.511843.... - Peter Luschny, Jan 05 2023

D-finite with recurrence 3*n*(3*n+2)*(3*n+1)*a(n) -8*(4*n-3)*(2*n-1)*(4*n-1)*a(n-1)=0. - R. J. Mathar, Jul 31 2024