A266336 - OEIS

%I #20 Sep 08 2022 08:46:15

%S 1,4,9,16,26,42,67,104,158,238,359,542,816,1224,1833,2746,4116,6168,

%T 9237,13828,20702,30998,46415,69492,104034,155746,233171,349090,

%U 522628,782420,1171349,1753622,2625352,3930412,5884193,8809176,13188162,19743938,29558555

%N G.f. = b(2)*b(6)/(x^6-x^4+x^2-2*x+1), where b(k) = (1-x^k)/(1-x).

%C This is the Poincaré series [or Poincare series] for the quasi-Lannér diagram QL4_5 - see Tables 7.6, 7.7 and 7.8 in Maxim Chapovalov, Dimitry Leites and Rafael Stekolshchik (2009), or equivalently Tables 5 and 6 in the shorter version, Maxim Chapovalov, Dimitry Leites and Rafael Stekolshchik (2010).

%H Colin Barker, <a href="/A266336/b266336.txt">Table of n, a(n) for n = 0..1000</a>

%H Maxim Chapovalov, Dimitry Leites, and Rafael Stekolshchik, <a href="https://arxiv.org/abs/0906.1596">The Poincaré series [or Poincare 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 [or Poincare series] of the hyperbolic Coxeter groups with finite volume of fundamental domains</a>, Journal of Nonlinear Mathematical Physics 17.supp01 (2010), 169-215.

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,0,1,0,-1).

%p gf:= b(2)*b(6)/(x^6-x^4+x^2-2*x+1):

%p b:= k->(1-x^k)/(1-x):

%p a:= n-> coeff(series(gf, x, n+1), x, n):

%p seq(a(n), n=0..40);

%t b[k_] := (1 - x^k)/(1 - x); CoefficientList[Series[b[2] b[6]/(x^6 - x^4 + x^2 - 2 x + 1), {x, 0, 40}], x] (* _Bruno Berselli_, Dec 29 2015 *)

%o (Magma) /* By definition: */ m:=40; R<x>:=PowerSeriesRing(Integers(), m); b:=func<k|(1-x^k)/(1-x)>; Coefficients(R!(b(2)*b(6)/(x^6-x^4+x^2-2*x+1))); // _Bruno Berselli_, Dec 29 2015

%Y Cf. similar sequences listed in A265055.

%K nonn,easy

%O 0,2

%A _Alois P. Heinz_, Dec 27 2015