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%I #23 Sep 20 2022 19:04:19
%S 1,4,9,17,29,46,70,103,148,210,295,411,569,783,1074,1470,2008,2740,
%T 3736,5091,6934,9440,12848,17483,23786,32358,44016,59871,81435,110762,
%U 150646,204888,278657,378983,515426,700988,953353,1296570,1763345,2398159,3261505,4435655,6032499,8204206,11157727
%N Poincaré series for hyperbolic reflection group with Coxeter diagram o-(4)-o---o-(5)-o.
%H Colin Barker, <a href="/A265047/b265047.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 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 of the hyperbolic Coxeter groups with finite volume of fundamental domains</a>, Journal of Nonlinear Mathematical Physics 17.supp01 (2010): 169-215.
%H R. L. Worthington, <a href="http://dx.doi.org/10.4153/CMB-1998-033-5">The growth series of compact hyperbolic Coxeter groups, with 4 and 5 generators</a>, Canad. Math. Bull. 41(2) (1998) 231-239.
%H <a href="/index/Rec#order_15">Index entries for linear recurrences with constant coefficients</a>, signature (2,-2,2,-1,1,-1,1,-1,1,-1,1,-2,2,-2,1).
%F G.f.: -b(4)*b(5)*(x^3+1)*(x^5+1)/t1 where b(k) = (1-x^k)/(1-x) and t1=(x-1)*(x^6+x^3+1)*(x^8-x^7+x^6-2*x^5+x^4-2*x^3+x^2-x+1).
%F G.f.: (1 +x)^3*(1 -x +x^2)*(1 +x^2)*(1 -x +x^2 -x^3 +x^4)*(1 +x +x^2 +x^3 +x^4) / ((1 -x)*(1 +x^3 +x^6)*(1 -x +x^2 -2*x^3 +x^4 -2*x^5 +x^6 -x^7 +x^8)). - _Colin Barker_, Jan 01 2016
%p b:=n->(1-x^n)/(1-x);
%p t1:=(x-1)*(x^6+x^3+1)*(x^8-x^7+x^6-2*x^5+x^4-2*x^3+x^2-x+1);
%p t2:=-b(4)*b(5)*(x^3+1)*(x^5+1)/t1;
%p t3:=series(t2,x,50);
%p t4:=seriestolist(t3);
%t LinearRecurrence[{2,-2,2,-1,1,-1,1,-1,1,-1,1,-2,2,-2,1},{1,4,9,17,29,46,70,103,148,210,295,411,569,783,1074,1470},50] (* _Harvey P. Dale_, Sep 20 2022 *)
%o (PARI) Vec((1 +x)^3*(1 -x +x^2)*(1 +x^2)*(1 -x +x^2 -x^3 +x^4)*(1 +x +x^2 +x^3 +x^4) / ((1 -x)*(1 +x^3 +x^6)*(1 -x +x^2 -2*x^3 +x^4 -2*x^5 +x^6 -x^7 +x^8)) + O(x^50)) \\ _Colin Barker_, Jan 01 2016
%Y Poincaré series in this family: A265044 and A265047 - A265054.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_, Dec 27 2015
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