A265050 Poincaré series for hyperbolic reflection group with Coxeter diagram shown in Comments.

1, 4, 10, 21, 39, 68, 114, 186, 298, 472, 743, 1165, 1822, 2844, 4434, 6908, 10758, 16749, 26071, 40576, 63146, 98266, 152914, 237948, 370263, 576149, 896514, 1395012, 2170690, 3377668, 5255762, 8178133, 12725431, 19801164, 30811218, 47943194, 74601066, 116081520, 180626359, 281060077

Offset

0,2

Comments

The Coxeter diagram is:

o---o

|...|

|...| 4

|...|

o---o

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

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 (2,0,-2,2,0,-2,1).

Formula

G.f.: -b(2)*b(4)*(x^3+1)/t1 where b(k) = (1-x^k)/(1-x) and t1 = (x-1)*(x^6-x^5-x^4+x^3-x^2-x+1).

G.f.: (1+x)^3*(1-x+x^2)*(1+x^2) / ((1-x)*(1-x-x^2+x^3-x^4-x^5+x^6)). - Colin Barker, Jan 01 2016

Mathematica

Join[{1}, LinearRecurrence[{2, 0, -2, 2, 0, -2, 1}, {4, 10, 21, 39, 68, 114, 186}, 40]] (* Jean-François Alcover, Jan 07 2019 *)

Prog

(PARI) Vec((1+x)^3*(1-x+x^2)*(1+x^2)/((1-x)*(1-x-x^2+x^3-x^4-x^5+x^6)) + O(x^50)) \\ Colin Barker, Jan 01 2016

Crossrefs

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

Sequence in context: A325951 A023538 A085360 * A243738 A258352 A024988

Adjacent sequences: A265047 A265048 A265049 * A265051 A265052 A265053

Keyword

nonn,easy

Author

N. J. A. Sloane, Dec 27 2015

Status

approved