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%I #19 Sep 08 2022 08:46:15
%S 1,4,10,21,40,73,129,224,385,658,1122,1910,3248,5519,9375,15922,27038,
%T 45911,77954,132358,224727,381555,647823,1099903,1867461,3170650,
%U 5383253,9139893,15518057,26347142,44733168,75949650,128950161,218936410,371718429,631117454
%N Expansion of b(2)*b(4)*b(6)/(1-x-x^2-x^4+x^6+x^8), where b(k) = (1-x^k)/(1-x).
%C This is the Poincaré series [or Poincare series] for the quasi-Lannér diagram QL4_18 - see Table 7.8 in Maxim Chapovalov, Dimitry Leites and Rafael Stekolshchik (2009), or equivalently Table 6 in the shorter version, Maxim Chapovalov, Dimitry Leites and Rafael Stekolshchik (2010).
%H Colin Barker, <a href="/A266355/b266355.txt">Table of n, a(n) for n = 0..1000</a>
%H Maxim Chapovalov, Dimitry Leites, and Rafael Stekolshchik, <a href="http://arxiv.org/abs/0906.1596">The Poincaré series [or Poincare series] of the hyperbolic Coxeter groups with finite volume of fundamental domains</a>, arXiv:0906.1596 [math.RT], 2009, page 31.
%H Maxim Chapovalov, Dimitry Leites, and Rafael Stekolshchik, <a href="http://dx.doi.org/10.1142/S1402925110000842">The Poincaré series [or Poincare series] of the hyperbolic Coxeter groups with finite volume of fundamental domains</a>, Journal of Nonlinear Mathematical Physics, Volume 17, Supplement 1 (2010), page 186.
%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,0,1,0,-1,0,-1).
%F G.f.: (1 + x^2)*(1 - x + x^2)*(1 + x + x^2)*(1 + x)^3/((1 - x)*(1 - x^2 - x^3 - 2*x^4 - 2*x^5 - x^6 - x^7)).
%F a(n) = a(n-1) + a(n-2) + a(n-4) - a(n-6) - a(n-8) for n>9.
%t CoefficientList[Series[(1 + x^2) (1 - x + x^2) (1 + x + x^2) (1 + x)^3/((1 - x) (1 - x^2 - x^3 - 2 x^4 - 2 x^5 - x^6 - x^7)), {x, 0, 40}], x]
%t LinearRecurrence[{1,1,0,1,0,-1,0,-1},{1,4,10,21,40,73,129,224,385,658},40] (* _Harvey P. Dale_, Apr 04 2019 *)
%o (Magma) /* By definition: */ m:=40; R<x>:=PowerSeriesRing(Integers(), m); b:=func<k|(1-x^k)/(1-x)>; Coefficients(R!(b(2)*b(4)*b(6)/(1-x-x^2-x^4+x^6+x^8)));
%Y Cf. similar sequences listed in A265055.
%K nonn,easy
%O 0,2
%A _Bruno Berselli_, Dec 28 2015
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