A265044 Poincaré series for hyperbolic reflection group with Coxeter diagram o---o-(5)-o---o.

1, 4, 9, 16, 26, 41, 62, 90, 128, 181, 254, 352, 483, 660, 900, 1224, 1661, 2252, 3052, 4133, 5592, 7562, 10224, 13821, 18680, 25244, 34113, 46096, 62284, 84152, 113695, 153608, 207530, 280377, 378792, 511750, 691374, 934041, 1261878, 1704780, 2303133, 3111496, 4203578, 5678960, 7672173, 10364964

Offset

0,2

Comments

The diagram is o---o-(5)-o---o.

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

Formula

G.f.: -b(2)*b(3)*(1+x^3)*(1+x^5)/((x-1)*(x^10-x^9-x^6+x^5-x^4-x+1)) where b(k) = (1-x^k)/(1-x).

G.f.: (1+x)^3*(1-x+x^2)*(1+x+x^2)*(1-x+x^2-x^3+x^4) / ((1-x)*(1-x-x^4+x^5-x^6-x^9+x^10)). - Colin Barker, Jan 01 2016

Prog

(PARI) Vec((1+x)^3*(1-x+x^2)*(1+x+x^2)*(1-x+x^2-x^3+x^4)/((1-x)*(1-x-x^4+x^5-x^6-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: A006508 A299898 A009875 * A266336 A027365 A100216

Adjacent sequences: A265041 A265042 A265043 * A265045 A265046 A265047

Keyword

nonn,easy

Author

N. J. A. Sloane, Dec 27 2015

Status

approved