A265049 Poincaré series for hyperbolic reflection group with Coxeter diagram shown in Comments.

1, 4, 9, 17, 29, 47, 74, 113, 170, 253, 374, 550, 804, 1171, 1702, 2472, 3588, 5204, 7545, 10936, 15848, 22962, 33265, 48188, 69803, 101112, 146461, 212145, 307283, 445083, 644676, 933770, 1352500, 1958999, 2837467, 4109861, 5952825, 8622216, 12488623, 18088816, 26200265, 37949074, 54966320, 79614492

Offset

0,2

Comments

The Coxeter diagram is:

o

|

5|

|

o---o---o

(4 nodes, T-shaped, 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,-4,4,-3,2,-2,3,-4,4,-3,1).

Formula

G.f.: (t^2 + 1)*(t + 1)*(t^3 + 1)*(t^5 + 1)/t1 where t1 = (t-1)*(t^10 - 2*t^9 + 2*t^8 - 2*t^7 + t^6 - t^5 + t^4 - 2*t^3 + 2*t^2 - 2*t + 1).

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+2*x^2-2*x^3+x^4-x^5+x^6-2*x^7+2*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: A265047 A266338 A301124 * A266333 A008225 A057313

Adjacent sequences: A265046 A265047 A265048 * A265050 A265051 A265052

Keyword

nonn,easy

Author

N. J. A. Sloane, Dec 27 2015

Status

approved