A265054 Poincaré series for hyperbolic reflection group with Coxeter diagram shown in Comments.

1, 4, 10, 22, 46, 94, 188, 372, 734, 1446, 2844, 5588, 10976, 21556, 42330, 83120, 163214, 320484, 629292, 1235652, 2426272, 4764118, 9354602, 18368260, 36067056, 70819582, 139058010, 273047782, 536143806, 1052746804, 2067124190, 4058907988, 7969881118, 15649284294, 30728199738, 60336449982

Offset

0,2

Comments

The Coxeter diagram is:

..5

o---o

|...|

|...|

|...|

o---o

..5

(4 nodes, square, two opposite edges carry 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,-2,-1,3,-3,1,2,-3,1).

Formula

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

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

Prog

(PARI) Vec((1+x)^3*(1-x+x^2)*(1-x+x^2-x^3+x^4)/((1-x)*(1+x^2)*(1-2*x-x^2+3*x^3-x^4-2*x^5+x^6)) + O(x^40)) \\ Colin Barker, Jan 01 2016

Crossrefs

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

Sequence in context: A078407 A347113 A347307 * A099018 A033484 A296953

Adjacent sequences: A265051 A265052 A265053 * A265055 A265056 A265057

Keyword

nonn,easy

Author

N. J. A. Sloane, Dec 27 2015

Status

approved