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A265047
Poincaré 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
OFFSET
0,2
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.
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);
MATHEMATICA
LinearRecurrence[{2, -2, 2, -1, 1, -1, 1, -1, 1, -1, 1, -2, 2, -2, 1}, {1, 4, 9, 17, 29, 46, 70, 103, 148, 210, 295, 411, 569, 783, 1074, 1470}, 50] (* Harvey P. Dale, Sep 20 2022 *)
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
Poincaré series in this family: A265044 and A265047 - A265054.
Sequence in context: A181286 A008138 A301123 * A266338 A301124 A265049
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, Dec 27 2015
STATUS
approved