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A045720 3-fold convolution of A001700(n), n >= 0. 6
1, 9, 57, 312, 1578, 7599, 35401, 161052, 719790, 3173090, 13836426, 59803104, 256596276, 1094249019, 4642178601, 19605872724, 82483419846, 345839048094, 1445715336366, 6027524015664, 25070662980876, 104056307673654 (list; graph; refs; listen; history; text; internal format)
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
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
Sequence in context: A064838 A027210 A192054 * A014916 A045635 A026896
KEYWORD
easy,nonn
AUTHOR
STATUS
approved

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Last modified February 26 12:49 EST 2024. Contains 370352 sequences. (Running on oeis4.)