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%I #58 Mar 25 2023 15:29:57
%S 1,9,57,312,1578,7599,35401,161052,719790,3173090,13836426,59803104,
%T 256596276,1094249019,4642178601,19605872724,82483419846,345839048094,
%U 1445715336366,6027524015664,25070662980876,104056307673654
%N 3-fold convolution of A001700(n), n >= 0.
%C Total number of 132 (or 213) patterns in the set of all 123-avoiding permutations of length (n+3). - _Cheyne Homberger_, Mar 16 2012
%C a(n) is the degree of the cyclic graphical Gaussian model for the (n+3) cycle. - _Mateusz Michalek_, Mar 04 2023
%D 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"
%H Indranil Ghosh, <a href="/A045720/b045720.txt">Table of n, a(n) for n = 0..1500</a>
%H José Agapito, Ângela Mestre, Maria M. Torres, and Pasquale Petrullo, <a href="http://cs.uwaterloo.ca/journals/JIS/VOL18/Agapito/agapito2.html">On One-Parameter Catalan Arrays</a>, Journal of Integer Sequences, Vol. 18 (2015), Article 15.5.1.
%H A. Ayyer, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL12/Ayyer/ayyer7.html">Towards a Human Proof of Gessel's Conjecture</a>, JIS 12 (2009) 09.4.2
%H R. Dinu, M. Michalek, and M. Vodička. <a href="https://arxiv.org/abs/2111.02937">Geometry of the Gaussian graphical model of the cycle</a>, arXiv preprint arXiv:2111.02937 [math.AG] (2021).
%H C. Homberger, <a href="https://doi.org/10.37236/2515">Expected patterns in permutation classes</a>, Electronic Journal of Combinatorics, 19(3) (2012), P43.
%H Milan Janjić, <a href="https://www.emis.de/journals/JIS/VOL21/Janjic2/janjic103.html">Pascal Matrices and Restricted Words</a>, J. Int. Seq., Vol. 21 (2018), Article 18.5.2.
%H D. R. Snow, <a href="http://www.jstor.org/stable/2686271">Spreadsheets, Power Series, Generating Functions and Integers</a>, The College Maths. J. 20 (1989) 149.
%F a(n) = (n+5)*binomial(2*(n+3), n+3)/4 - 3*2^(2*n+3);
%F G.f.: (c(x)/sqrt(1-4*x))^3, where c(x) = g.f. for Catalan numbers A000108;
%F recursion: a(n)=(2*(2*n+7)/(n+3))*a(n-1)+(3/(n+3))*A008549(n+1), a(0)=1.
%t Table[(n+5)*Binomial[2*(n+3),n+3]/4-3*2^(2n+3),{n,0,21}] (* _Indranil Ghosh_, Feb 18 2017 *)
%o (Python)
%o import math
%o def C(n,r):
%o ....f=math.factorial
%o ....return f(n)/f(r)/f(n-r)
%o def A045720(n):
%o ....return (n+5)*C(2*(n+3),n+3)/4-3*2**(2*n+3) # _Indranil Ghosh_, Feb 18 2017
%o (PARI) x='x+O('x^30); Vec((((1-4*x)^(-1/2)-1)/(2*x))^3) \\ _Altug Alkan_, Sep 04 2018
%Y Cf. A000108, A001700, A008549.
%K easy,nonn
%O 0,2
%A _Wolfdieter Lang_
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