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A265047 Poincare series for hyperbolic reflection group with Coxeter diagram o-(4)-o---o-(5)-o. 9
1, 4, 9, 17, 29, 46, 70, 103, 148, 210, 295, 411, 569, 783, 1074, 1470, 2008, 2740, 3736, 5091, 6934, 9440, 12848, 17483, 23786, 32358, 44016, 59871, 81435, 110762, 150646, 204888, 278657, 378983, 515426, 700988, 953353, 1296570, 1763345, 2398159, 3261505, 4435655, 6032499, 8204206, 11157727 (list; graph; refs; listen; history; text; internal format)
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 [or Poincare 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 [or Poincare 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,-2,2,-1,1,-1,1,-1,1,-1,1,-2,2,-2,1).

FORMULA

G.f.: -b(4)*b(5)*(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^8-x^7+x^6-2*x^5+x^4-2*x^3+x^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 +x^2 +x^3 +x^4) / ((1 -x)*(1 +x^3 +x^6)*(1 -x +x^2 -2*x^3 +x^4 -2*x^5 +x^6 -x^7 +x^8)). - Colin Barker, Jan 01 2016

MAPLE

b:=n->(1-x^n)/(1-x);

t1:=(x-1)*(x^6+x^3+1)*(x^8-x^7+x^6-2*x^5+x^4-2*x^3+x^2-x+1);

t2:=-b(4)*b(5)*(x^3+1)*(x^5+1)/t1;

t3:=series(t2, x, 50);

t4:=seriestolist(t3);

PROG

(PARI) Vec((1 +x)^3*(1 -x +x^2)*(1 +x^2)*(1 -x +x^2 -x^3 +x^4)*(1 +x +x^2 +x^3 +x^4) / ((1 -x)*(1 +x^3 +x^6)*(1 -x +x^2 -2*x^3 +x^4 -2*x^5 +x^6 -x^7 +x^8)) + O(x^50)) \\ Colin Barker, Jan 01 2016

CROSSREFS

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

Sequence in context: A181286 A008138 A301123 * A266338 A301124 A265049

Adjacent sequences:  A265044 A265045 A265046 * A265048 A265049 A265050

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane, Dec 27 2015

STATUS

approved

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Last modified September 15 20:42 EDT 2019. Contains 327087 sequences. (Running on oeis4.)