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%I #14 Sep 08 2022 08:46:02
%S 1,1,2,5,14,30,72,195,485,1059,2065,3682,6120,9620,14454,20925,29367,
%T 40145,53655,70324,90610,115002,144020,178215,218169,264495,317837,
%U 378870,448300,526864,615330,714497,825195,948285,1084659,1235240,1400982,1582870,1781920
%N Expansion of (1-4*x+7*x^2-5*x^3+4*x^4-6*x^5+21*x^6+18*x^7-5*x^8)/(1-x)^5.
%C In the paper of Kitaev, Remmel and Tiefenbruck (see the Links section), Q_(132)^(0,k,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, the g.f. of A028310: (1-x+x^2)/(1-x)^2;
%C k=3, the g.f. (1-2*x+2*x^2+x^3-x^4)/(1-x)^3, whose coefficients (except the first two) are given by A000096 (for n>0);
%C k=4, the g.f. (1-3*x+4*x^2-x^3+3*x^4-5*x^5+2*x^6)/(1-x)^4, whose coefficients (except the first three) are given by A005586 (for n>0).
%C This sequence corresponds to the case k=5.
%H Bruno Berselli, <a href="/A212393/b212393.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 21, Theorem 10).
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F G.f.: (1-4*x+7*x^2-5*x^3+4*x^4-6*x^5+21*x^6+18*x^7-5*x^8)/(1-x)^5.
%F a(n) = 5*a(n-1)-10*a(n-2)+10*a(n-3)-5*a(n-4)+a(n-5) for n>8, a(0)=a(1)=1, a(2)=2, a(3)=5, a(4)=14, a(5)=30, a(6)=72, a(7)=195, a(8)=485.
%F a(n) = (n-3)*(31*n^3-369*n^2+1454*n-1560)/24 for n>3, a(0)=a(1)=1, a(2)=2, a(3)=5.
%F G.f.: 1+x+2*x^2+5*x^3 + 14*x^4*G(0), where G(k)= 1 + x*(k+1)*(124*k^3+192*k^2+89*k+180)/( (2*k+1)*(62*k^3+3*k^2-5*k+84) - x*(2*k+1)*(62*k^3+3*k^2-5*k+84)*(2*k+3)*(62*k^3+189*k^2+187*k+144)/(x*(2*k+3)*(62*k^3+189*k^2+187*k+144) + (k+1)*(124*k^3+192*k^2+89*k+180)/G(k+1) )); (continued fraction). - _Sergei N. Gladkovskii_, Jul 06 2013
%t CoefficientList[Series[(1 - 4 x + 7 x^2 - 5 x^3 + 4 x^4 - 6 x^5 + 21 x^6 + 18 x^7 - 5 x^8)/(1 - x)^5, {x, 0, 38}], x]
%o (PARI) Vec((1-4*x+7*x^2-5*x^3+4*x^4-6*x^5+21*x^6+18*x^7-5*x^8)/(1-x)^5+O(x^39))
%o (Magma) m:=39; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!((1-4*x+7*x^2-5*x^3+4*x^4-6*x^5+21*x^6+18*x^7-5*x^8)/(1-x)^5));
%K nonn,easy
%O 0,3
%A _Bruno Berselli_, May 14 2012
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