A265053 Poincaré series for hyperbolic reflection group with Coxeter diagram shown in Comments.

1, 4, 10, 21, 40, 73, 130, 228, 396, 684, 1178, 2025, 3476, 5961, 10218, 17512, 30010, 51424, 88114, 150977, 258684, 443225, 759410, 1301148, 2229340, 3819668, 6544474, 11213049, 19212000, 32917085, 56398834, 96631532, 165564642, 283671900, 486032194, 832748301, 1426797936, 2444619033

Offset

0,2

Comments

The Coxeter diagram is:

o---o

|...|

|...| 5

|...|

o---o

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

Links

Colin Barker, Table of n, a(n) for n = 0..1000

Maxim Chapovalov, Dimitry Leites, and Rafael Stekolshchik, The Poincaré series of the hyperbolic Coxeter groups with finite volume of fundamental domains, arXiv:0906.1596 [math.RT], 2009.

Maxim Chapovalov, Dimitry Leites, and Rafael Stekolshchik, The Poincaré series of the hyperbolic Coxeter groups with finite volume of fundamental domains, Journal of Nonlinear Mathematical Physics 17.supp01 (2010): 169-215.

R. L. Worthington, The growth series of compact hyperbolic Coxeter groups, with 4 and 5 generators, Canad. Math. Bull. 41(2) (1998) 231-239

Index entries for linear recurrences with constant coefficients, signature (3,-3,1,2,-4,4,-2,-1,3,-3,1).

Formula

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).

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

Mathematica

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 *)

Prog

(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

Crossrefs

Poincaré series in this family: A265044 and A265047 - A265054.

Sequence in context: A144897 A001891 A266355 * A266371 A293823 A266354

Adjacent sequences: A265050 A265051 A265052 * A265054 A265055 A265056

Keyword

nonn,easy

Author

N. J. A. Sloane, Dec 27 2015

Status

approved