|
%I #39 Jan 06 2023 06:53:13
%S 5,2,5,20,100,570,3542,23400,161820,1159400,8544965,64448228,
%T 495508780,3872033900,30680401500,246041115600,1993987498284,
%U 16310419381080,134519771966180,1117653277802000,9347742311507600,78652006531467930,665393840873409150,5657273782416664200,48318619683648190500
%N a(n) = (10 / ((3*n+1)*(3*n+2))) * binomial(4*n, n).
%C A combinatorial interpretation for this sequence in terms of a family of plane trees is given in [Schaeffer, Corollary 2 with k = 4].
%C 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
%H Michael De Vlieger, <a href="/A197271/b197271.txt">Table of n, a(n) for n = 0..1031</a>
%H William G. Brown, <a href="http://dx.doi.org/10.1112/plms/s3-14.4.746">Enumeration of Triangulations of the Disk</a>, Proc. Lond. Math. Soc. s3-14 (1964) 746-768.
%H W. G. Brown, <a href="/A002709/a002709.pdf">Enumeration of Triangulations of the Disk</a>, Proc. Lond. Math. Soc. s3-14 (1964) 746-768. [Annotated scanned copy]
%H K. A. Penson, K. Górska, A. Horzela, and G. H. E. Duchamp, <a href="https://arxiv.org/abs/2209.06574">Hausdorff moment problem for combinatorial numbers of Brown and Tutte: exact solution</a>, arXiv:2209.06574 [math.CO], 2022.
%H G. Schaeffer, <a href="http://www.lix.polytechnique.fr/~schaeffe/Biblio/Sc03.ps">A combinatorial interpretation of super-Catalan numbers of order two</a>, (2001).
%H William T. Tutte, <a href="https://doi.org/10.1016/0095-8956(80)90059-3">On the enumeration of convex polyhedra</a>, J. Combin. Theory Ser. B 28 (1980), 105-126.
%F a(n) = 10/((3*n+1)*(3*n+2))*binomial(4*n,n).
%F a(n) = A000260(n) * 5*(n+1)/(4*n+1). - _Noam Zeilberger_, May 20 2019
%F a(n) ~ c*(256/27)^n / n^(5/2), where c = (10/9)*sqrt(2/(3*Pi)) = 0.511843.... - _Peter Luschny_, Jan 05 2023
%t Table[10/((3n+1)(3n+2)) Binomial[4n,n],{n,0,30}] (* _Harvey P. Dale_, Jan 27 2015 *)
%Y Column m=1 of A146305.
%Y Cf. A000139, A000260, A197272.
%K nonn,easy
%O 0,1
%A _Peter Bala_, Oct 12 2011
|
