|
|
A197271
|
|
a(n) = (10 / ((3*n+1)*(3*n+2))) * binomial(4*n, n).
|
|
6
|
|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
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 = 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
|
|
|
FORMULA
|
a(n) = 10/((3*n+1)*(3*n+2))*binomial(4*n,n).
a(n) ~ c*(256/27)^n / n^(5/2), where c = (10/9)*sqrt(2/(3*Pi)) = 0.511843.... - Peter Luschny, Jan 05 2023
|
|
MATHEMATICA
|
Table[10/((3n+1)(3n+2)) Binomial[4n, n], {n, 0, 30}] (* Harvey P. Dale, Jan 27 2015 *)
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn,easy
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|