A265052 Poincaré series for hyperbolic reflection group with Coxeter diagram shown in Comments.

1, 4, 10, 22, 45, 89, 172, 328, 622, 1176, 2220, 4186, 7888, 14859, 27987, 52710, 99268, 186946, 352062, 663010, 1248588, 2351350, 4428075, 8338971, 15703984, 29573802, 55693492, 104882184, 197514502, 371960008, 700476396, 1319139609, 2484208341, 4678269862, 8810134210, 16591275636, 31244748450

Offset

0,2

Comments

The Coxeter diagram is:

..5

o---o

|...|

|...|

|...|

o---o

..4

(4 nodes, square, a pair of opposite edges carry labels 4 and 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,-3,2,0,-2,2,0,-2,3,-3,1).

Formula

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

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

a(n) = 3*a(n-1) - 3*a(n-2) + 2*a(n-3) - 2*a(n-5) + 2*a(n-6) - 2*a(n-8) + 3*a(n-9) - 3*a(n-10) + a(n-11) for n>11. - Vincenzo Librandi, Jan 01 2016

Mathematica

Join[{1}, LinearRecurrence[{3, -3, 2, 0, -2, 2, 0, -2, 3, -3, 1}, {4, 10, 22, 45, 89, 172, 328, 622, 1176, 2220, 4186}, 60]] (* Vincenzo Librandi, Jan 01 2016 *)

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

Crossrefs

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

Sequence in context: A265051 A266375 A004798 * A266372 A174622 A038621

Adjacent sequences: A265049 A265050 A265051 * A265053 A265054 A265055

Keyword

nonn,easy

Author

N. J. A. Sloane, Dec 27 2015

Status

approved