%I #19 Aug 14 2020 17:30:50
%S 1,4,9,17,30,50,80,125,193,296,450,680,1025,1541,2312,3466,5194,7781,
%T 11653,17448,26122,39104,58533,87613,131138,196282,293784,439717,
%U 658137,985048,1474338,2206664,3302745,4943261,7398640,11073634,16574038,24806553,37128249,55570268,83172642,124485420
%N Poincaré series for hyperbolic reflection group with Coxeter diagram o-(5)-o---o-(5)-o.
%H Colin Barker, <a href="/A265048/b265048.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_13">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,0,1,-1,1,-1,1,-1,0,1,-2,1).
%F G.f.: -b(2)*b(5)*(x^3+1)*(x^5+1)/t1 where b(k) = (1-x^k)/(1-x) and t1=(x-1)*(x^12-x^11-x^8-x^6-x^4-x+1).
%F G.f.: (1+x)^3*(1-x+x^2)*(1-x+x^2-x^3+x^4)*(1+x+x^2+x^3+x^4) / ((1-x)*(1-x-x^4-x^6-x^8-x^11+x^12)). - _Colin Barker_, Jan 01 2016
%t LinearRecurrence[{2,-1,0,1,-1,1,-1,1,-1,0,1,-2,1},{1,4,9,17,30,50,80,125,193,296,450,680,1025,1541},50] (* _Harvey P. Dale_, Nov 25 2017 *)
%o (PARI) Vec((1+x)^3*(1-x+x^2)*(1-x+x^2-x^3+x^4)*(1+x+x^2+x^3+x^4)/((1-x)*(1-x-x^4-x^6-x^8-x^11+x^12)) + 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