A265053 - OEIS

%I #18 Aug 14 2020 17:30:33

%S 1,4,10,21,40,73,130,228,396,684,1178,2025,3476,5961,10218,17512,

%T 30010,51424,88114,150977,258684,443225,759410,1301148,2229340,

%U 3819668,6544474,11213049,19212000,32917085,56398834,96631532,165564642,283671900,486032194,832748301,1426797936,2444619033

%N Poincaré series for hyperbolic reflection group with Coxeter diagram shown in Comments.

%C The Coxeter diagram is:

%C o---o

%C |...|

%C |...| 5

%C |...|

%C o---o

%C (4 nodes, square, one edge carries label 5)

%H Colin Barker, <a href="/A265053/b265053.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_11">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1,2,-4,4,-2,-1,3,-3,1).

%F G.f.: -b(4)*(x^3+1)*(x^5+1)/t1 where b(k) = (1-x^k)/(1-x) and t1 = (x-1)*(x^10 - 2*x^9 + x^8 - 2*x^6 + 2*x^5  - 2*x^4 + x^2 - 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)*(1-2*x+x^2-2*x^4+2*x^5-2*x^6+x^8-2*x^9+x^10)). - _Colin Barker_, Jan 01 2016

%t Join[{1}, LinearRecurrence[{3, -3, 1, 2, -4, 4, -2, -1, 3, -3, 1}, {4, 10, 21, 40, 73, 130, 228, 396, 684, 1178, 2025}, 60]] (* _Vincenzo Librandi_, Jan 01 2016 *)

%o (PARI) Vec((1+x)^3*(1-x+x^2)*(1+x^2)*(1-x+x^2-x^3+x^4) / ((1-x)*(1-2*x+x^2-2*x^4+2*x^5-2*x^6+x^8-2*x^9+x^10)) + 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