A265048 Poincaré series for hyperbolic reflection group with Coxeter diagram o-(5)-o---o-(5)-o.

1, 4, 9, 17, 30, 50, 80, 125, 193, 296, 450, 680, 1025, 1541, 2312, 3466, 5194, 7781, 11653, 17448, 26122, 39104, 58533, 87613, 131138, 196282, 293784, 439717, 658137, 985048, 1474338, 2206664, 3302745, 4943261, 7398640, 11073634, 16574038, 24806553, 37128249, 55570268, 83172642, 124485420

Offset

0,2

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

Formula

G.f.: -b(2)*b(5)*(x^3+1)*(x^5+1)/t1 where b(k) = (1-x^k)/(1-x) and t1=(x-1)*(x^12-x^11-x^8-x^6-x^4-x+1).

G.f.: (1+x)^3*(1-x+x^2)*(1-x+x^2-x^3+x^4)*(1+x+x^2+x^3+x^4) / ((1-x)*(1-x-x^4-x^6-x^8-x^11+x^12)). - Colin Barker, Jan 01 2016

Mathematica

LinearRecurrence[{2, -1, 0, 1, -1, 1, -1, 1, -1, 0, 1, -2, 1}, {1, 4, 9, 17, 30, 50, 80, 125, 193, 296, 450, 680, 1025, 1541}, 50] (* Harvey P. Dale, Nov 25 2017 *)

Prog

(PARI) Vec((1+x)^3*(1-x+x^2)*(1-x+x^2-x^3+x^4)*(1+x+x^2+x^3+x^4)/((1-x)*(1-x-x^4-x^6-x^8-x^11+x^12)) + O(x^50)) \\ Colin Barker, Jan 01 2016

Crossrefs

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

Sequence in context: A008046 A008093 A027374 * A266334 A157728 A266335

Adjacent sequences: A265045 A265046 A265047 * A265049 A265050 A265051

Keyword

nonn,easy

Author

N. J. A. Sloane, Dec 27 2015

Status

approved