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%I #19 Sep 08 2022 08:46:14
%S 1,4,10,22,44,84,156,284,512,918,1642,2932,5230,9324,16618,29614,
%T 52768,94020,167516,298460,531756,947406,1687946,3007324,5357986,
%U 9546028,17007626,30301534,53986540,96184780,171367004,305314932,543962400,969147134,1726674794,3076319100,5480904238
%N Poincaré series for hyperbolic reflection group with Coxeter diagram shown in Comments.
%C The Coxeter diagram is:
%C ..4
%C o---o
%C |...|
%C |...|
%C |...|
%C o---o
%C ..4
%C (4 nodes, square, two opposite edges carry label 4)
%H Colin Barker, <a href="/A265051/b265051.txt">Table of n, a(n) for n = 0..1000</a>
%H Maxim Chapovalov, Dimitry Leites, and Rafael Stekolshchik, <a href="http://arxiv.org/abs/0906.1596">The Poincaré series of the hyperbolic Coxeter groups with finite volume of fundamental domains</a>, arXiv:0906.1596 [math.RT], 2009.
%H Maxim Chapovalov, Dimitry Leites, and Rafael Stekolshchik, <a href="http://dx.doi.org/10.1142/S1402925110000842">The Poincaré series of the hyperbolic Coxeter groups with finite volume of fundamental domains</a>, Journal of Nonlinear Mathematical Physics 17.supp01 (2010): 169-215.
%H R. L. Worthington, <a href="http://dx.doi.org/10.4153/CMB-1998-033-5">The growth series of compact hyperbolic Coxeter groups, with 4 and 5 generators</a>, Canad. Math. Bull. 41(2) (1998) 231-239
%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,-1,1,0,-2,1).
%F 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^2-x+1).
%F G.f.: (1+x)^3*(1-x+x^2)*(1+x^2) / ((1-x)*(1-x-x^2-x^4-x^5+x^6)). - _Colin Barker_, Jan 01 2016
%t Join[{1}, LinearRecurrence[{2, 0, -1, 1, 0, -2, 1},{4, 10, 22, 44, 84, 156, 284}, 60]] (* _Vincenzo Librandi_, Jan 01 2016 *)
%o (PARI) Vec((1+x)^3*(1-x+x^2)*(1+x^2)/((1-x)*(1-x-x^2-x^4-x^5+x^6)) + O(x^50)) \\ _Colin Barker_, Jan 01 2016
%o (Magma) I:=[1,4,10,22,44,84,156,284]; [n le 8 select I[n] else 2*Self(n-1)-Self(n-3)+Self(n-4)-2*Self(n-6)+Self(n-7): n in [1..50]]; // _Vincenzo Librandi_, Jan 01 2016
%Y Poincaré series in this family: A265044 and A265047 - A265054.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_, Dec 27 2015
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