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A211216 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).

%I A211216 #72 Oct 26 2024 22:57:13
%S A211216 1,1,2,5,14,42,132,429,1430,4862,16795,58766,207783,740924,2660139,
%T A211216 9603089,34818270,126676726,462125928,1689438278,6186432967,
%U A211216 22682699779,83249302471,305773834030,1123771473120,4131947428007,15197952958467,55915691993228
%N A211216 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 A211216 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 A211216 For successive values of k we have:
%C A211216 k=1, the g.f. of A000012: 1/(1-x);
%C A211216 k=2,      "      A011782: (1-x)/(1-2*x);
%C A211216 k=3,      "      A001519: (1-2*x)/(1-3*x+x^2);
%C A211216 k=4,      "      A124302: (1-3*x+x^2)/(1-4*x+3*x^2);
%C A211216 k=5,      "      A080937: (1-4*x+3*x^2)/(1-5*x+6*x^2-x^3);
%C A211216 k=6,      "      A024175: (1-5*x+6*x^2-x^3)/(1-6*x+10*x^2-4*x^3);
%C A211216 k=7,      "      A080938: (1-6*x+10*x^2-4*x^3)/(1-7*x+15*x^2-10*x^3+x^4);
%C A211216 k=8,      "      A033191: (1-7*x+15*x^2-10*x^3+x^4)/(1-8*x+21*x^2
%C A211216                            -20*x^3+5*x^4).
%C A211216 This sequence corresponds to the case k=9.
%C A211216 We observe that the coefficients of numerators and denominators are in A115139.
%C A211216 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 A211216 Bruno Berselli, <a href="/A211216/b211216.txt">Table of n, a(n) for n = 0..1000</a>
%H A211216 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 A211216 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 A211216 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 A211216 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (9,-28,35,-15,1).
%F A211216 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 A211216 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 A211216 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 A211216 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 A211216 (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 A211216 (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 A211216 Cf. A000108, A001519, A011782, A024175, A033191, A080937, A080938, A124302.
%Y A211216 Cf. square array in A080934.
%K A211216 nonn,easy
%O A211216 0,3
%A A211216 _Bruno Berselli_, May 11 2012