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Expansion of (1-8*x+21*x^2-20*x^3+5*x^4)/(1-9*x+28*x^2-35*x^3+15*x^4-x^5).
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%I #72 Oct 26 2024 22:57:13

%S 1,1,2,5,14,42,132,429,1430,4862,16795,58766,207783,740924,2660139,

%T 9603089,34818270,126676726,462125928,1689438278,6186432967,

%U 22682699779,83249302471,305773834030,1123771473120,4131947428007,15197952958467,55915691993228

%N Expansion of (1-8*x+21*x^2-20*x^3+5*x^4)/(1-9*x+28*x^2-35*x^3+15*x^4-x^5).

%C In the paper of Kitaev, Remmel and Tiefenbruck (see the Links section), Q_(132)^(k,0,0,0)(x,0) represents a generating function depending on k and x.

%C For successive values of k we have:

%C k=1, the g.f. of A000012: 1/(1-x);

%C k=2, " A011782: (1-x)/(1-2*x);

%C k=3, " A001519: (1-2*x)/(1-3*x+x^2);

%C k=4, " A124302: (1-3*x+x^2)/(1-4*x+3*x^2);

%C k=5, " A080937: (1-4*x+3*x^2)/(1-5*x+6*x^2-x^3);

%C k=6, " A024175: (1-5*x+6*x^2-x^3)/(1-6*x+10*x^2-4*x^3);

%C k=7, " A080938: (1-6*x+10*x^2-4*x^3)/(1-7*x+15*x^2-10*x^3+x^4);

%C k=8, " A033191: (1-7*x+15*x^2-10*x^3+x^4)/(1-8*x+21*x^2

%C -20*x^3+5*x^4).

%C This sequence corresponds to the case k=9.

%C We observe that the coefficients of numerators and denominators are in A115139.

%C In general, Q_(132)^(k,0,0,0)(x,0) is the generating function for Dyck paths whose maximum height is less than or equal to k; also, it is the generating function of rooted binary trees T which have no nodes 'eta' such that there are >= k left edges on the path from 'eta' to the root of T (see cited paper, page 11).

%H Bruno Berselli, <a href="/A211216/b211216.txt">Table of n, a(n) for n = 0..1000</a>

%H Sergey Kitaev, Jeffrey Remmel and Mark Tiefenbruck, <a href="http://arxiv.org/abs/1201.6243">Marked mesh patterns in 132-avoiding permutations I,</a> arXiv:1201.6243v1 [math.CO], 2012 (page 10, Corollary 3).

%H K. Mészáros, A. H. Morales, and J. Striker, <a href="http://arxiv.org/abs/1510.03357">On flow polytopes, order polytopes, and certain faces of the alternating sign matrix polytope</a>, arXiv preprint arXiv:1510.03357 [math.CO], 2015-2019.

%H László Németh and László Szalay, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL24/Nemeth/nemeth8.html">Sequences Involving Square Zig-Zag Shapes</a>, J. Int. Seq., Vol. 24 (2021), Article 21.5.2.

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (9,-28,35,-15,1).

%F G.f.: (1-3*x+x^2)*(1-5*x+5*x^2)/(1-9*x+28*x^2-35*x^3+15*x^4-x^5).

%F G.f.: 1/(1-x/(1-x/(1-x/(1-x/(1-x/(1-x/(1-x/(1-x/(1-x))))))))). - _Philippe Deléham_, Mar 14 2013

%F a(n) = A000108(n) + Sum_{k=1..n} (4*binomial(2*n, n+11*k) - binomial(2*n+2, n+11*k+1)). - _Greg Dresden_, Jan 28 2023

%t CoefficientList[Series[(1 - 8 x + 21 x^2 - 20 x^3 + 5 x^4)/(1 - 9 x + 28 x^2 - 35 x^3 + 15 x^4 - x^5), {x, 0, 27}], x]

%o (PARI) Vec((1-8*x+21*x^2-20*x^3+5*x^4)/(1-9*x+28*x^2-35*x^3+15*x^4-x^5)+O(x^28))

%o (Magma) m:=28; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!((1-8*x+21*x^2-20*x^3+5*x^4)/(1-9*x+28*x^2-35*x^3+15*x^4-x^5)));

%Y Cf. A000108, A001519, A011782, A024175, A033191, A080937, A080938, A124302.

%Y Cf. square array in A080934.

%K nonn,easy,changed

%O 0,3

%A _Bruno Berselli_, May 11 2012