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%I #24 Sep 08 2022 08:46:15
%S 1,4,10,22,46,95,193,388,778,1558,3118,6236,12468,24926,49830,99614,
%T 199133,398073,795758,1590738,3179918,6356718,12707200,25401932,
%U 50778938,101508046,202916478,405633823,810869573,1620943388,3240296038,6477412158,12948467598
%N G.f. = b(2)*b(6)*b(10)/(x^15+x^14+x^13+x^12+x^11-2*x^5-x^4-x^3-x^2-x+1), where b(k) = (1-x^k)/(1-x).
%C This is the Poincaré series [or Poincare series] for the quasi-Lannér diagram QL4_12 - see Tables 7.6, 7.7 and 7.8 in Maxim Chapovalov, Dimitry Leites and Rafael Stekolshchik (2009), or equivalently Tables 5 and 6 in the shorter version, Maxim Chapovalov, Dimitry Leites and Rafael Stekolshchik (2010).
%H Colin Barker, <a href="/A266373/b266373.txt">Table of n, a(n) for n = 0..1000</a>
%H Maxim Chapovalov, Dimitry Leites, and Rafael Stekolshchik, <a href="https://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.
%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 17.supp01 (2010), 169-215.
%H <a href="/index/Rec#order_11">Index entries for linear recurrences with constant coefficients</a>, signature (2, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1).
%p gf:= b(2)*b(6)*b(10)/(x^15+x^14+x^13+x^12+x^11-2*x^5-x^4-x^3-x^2-x+1):
%p b:= k->(1-x^k)/(1-x):
%p a:= n-> coeff(series(gf, x, n+1), x, n):
%p seq(a(n), n=0..40);
%t b[k_] := (1 - x^k)/(1 - x); CoefficientList[Series[b[2] b[6] b[10]/(x^15 + x^14 + x^13 + x^12 + x^11 - 2 x^5 - x^4 - x^3 - x^2 - x + 1), {x, 0, 40}], x] (* _Bruno Berselli_, Dec 28 2015 *)
%t LinearRecurrence[{2,0,0,0,0,0,0,0,0,0,-1},{1,4,10,22,46,95,193,388,778,1558,3118,6236},40] (* _Harvey P. Dale_, Mar 14 2016 *)
%o (Magma) /* By definition: */ m:=40; R<x>:=PowerSeriesRing(Integers(), m); b:=func<k|(1-x^k)/(1-x)>; Coefficients(R!(b(2)*b(6)*b(10)/(x^15+x^14+x^13 +x^12+x^11-2*x^5-x^4-x^3-x^2-x+1))); // _Bruno Berselli_, Dec 29 2015
%Y Cf. similar sequences listed in A265055.
%K nonn,easy
%O 0,2
%A _Alois P. Heinz_, Dec 28 2015