A045720 3-fold convolution of A001700(n), n >= 0.

1, 9, 57, 312, 1578, 7599, 35401, 161052, 719790, 3173090, 13836426, 59803104, 256596276, 1094249019, 4642178601, 19605872724, 82483419846, 345839048094, 1445715336366, 6027524015664, 25070662980876, 104056307673654

Offset

0,2

Comments

Total number of 132 (or 213) patterns in the set of all 123-avoiding permutations of length (n+3). - Cheyne Homberger, Mar 16 2012

a(n) is the degree of the cyclic graphical Gaussian model for the (n+3) cycle. - Mateusz Michalek, Mar 04 2023

References

B. Sturmfels, and C. Uhler. Multivariate Gaussians, semidefinite matrix completion, and convex algebraic geometry. Annals of the Institute of Statistical Mathematics 62.4 (2010): 603-638, Conjecture 2 proved in "Geometry of the Gaussian graphical model of the cycle"

Links

Indranil Ghosh, Table of n, a(n) for n = 0..1500

José Agapito, Ângela Mestre, Maria M. Torres, and Pasquale Petrullo, On One-Parameter Catalan Arrays, Journal of Integer Sequences, Vol. 18 (2015), Article 15.5.1.

A. Ayyer, Towards a Human Proof of Gessel's Conjecture, JIS 12 (2009) 09.4.2

R. Dinu, M. Michalek, and M. Vodička. Geometry of the Gaussian graphical model of the cycle, arXiv preprint arXiv:2111.02937 [math.AG] (2021).

C. Homberger, Expected patterns in permutation classes, Electronic Journal of Combinatorics, 19(3) (2012), P43.

Milan Janjić, Pascal Matrices and Restricted Words, J. Int. Seq., Vol. 21 (2018), Article 18.5.2.

D. R. Snow, Spreadsheets, Power Series, Generating Functions and Integers, The College Maths. J. 20 (1989) 149.

Formula

a(n) = (n+5)*binomial(2*(n+3), n+3)/4 - 3*2^(2*n+3);

G.f.: (c(x)/sqrt(1-4*x))^3, where c(x) = g.f. for Catalan numbers A000108;

recursion: a(n)=(2*(2*n+7)/(n+3))*a(n-1)+(3/(n+3))*A008549(n+1), a(0)=1.

Mathematica

Table[(n+5)*Binomial[2*(n+3), n+3]/4-3*2^(2n+3), {n, 0, 21}] (* Indranil Ghosh, Feb 18 2017 *)

Prog

(Python)
import math
def C(n, r):
....f=math.factorial
....return f(n)/f(r)/f(n-r)
def A045720(n):
....return (n+5)*C(2*(n+3), n+3)/4-3*2**(2*n+3) # Indranil Ghosh, Feb 18 2017
(PARI) x='x+O('x^30); Vec((((1-4*x)^(-1/2)-1)/(2*x))^3) \\ Altug Alkan, Sep 04 2018

Crossrefs

Cf. A000108, A001700, A008549.

Sequence in context: A064838 A027210 A192054 * A014916 A045635 A026896

Adjacent sequences: A045717 A045718 A045719 * A045721 A045722 A045723

Keyword

easy,nonn

Author

Wolfdieter Lang

Status

approved