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A265050
Poincaré series for hyperbolic reflection group with Coxeter diagram shown in Comments.
1
1, 4, 10, 21, 39, 68, 114, 186, 298, 472, 743, 1165, 1822, 2844, 4434, 6908, 10758, 16749, 26071, 40576, 63146, 98266, 152914, 237948, 370263, 576149, 896514, 1395012, 2170690, 3377668, 5255762, 8178133, 12725431, 19801164, 30811218, 47943194, 74601066, 116081520, 180626359, 281060077
OFFSET
0,2
COMMENTS
The Coxeter diagram is:
o---o
|...|
|...| 4
|...|
o---o
(4 nodes, square, one edge carries label 4)
LINKS
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
FORMULA
G.f.: -b(2)*b(4)*(x^3+1)/t1 where b(k) = (1-x^k)/(1-x) and t1 = (x-1)*(x^6-x^5-x^4+x^3-x^2-x+1).
G.f.: (1+x)^3*(1-x+x^2)*(1+x^2) / ((1-x)*(1-x-x^2+x^3-x^4-x^5+x^6)). - Colin Barker, Jan 01 2016
MATHEMATICA
Join[{1}, LinearRecurrence[{2, 0, -2, 2, 0, -2, 1}, {4, 10, 21, 39, 68, 114, 186}, 40]] (* Jean-François Alcover, Jan 07 2019 *)
PROG
(PARI) Vec((1+x)^3*(1-x+x^2)*(1+x^2)/((1-x)*(1-x-x^2+x^3-x^4-x^5+x^6)) + O(x^50)) \\ Colin Barker, Jan 01 2016
CROSSREFS
Poincaré series in this family: A265044 and A265047 - A265054.
Sequence in context: A325951 A023538 A085360 * A376712 A243738 A258352
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, Dec 27 2015
STATUS
approved