A266333 - OEIS

%I #23 Sep 08 2022 08:46:15

%S 1,4,9,17,29,47,74,113,170,253,375,555,818,1203,1767,2594,3807,5584,

%T 8188,12004,17597,25795,37809,55416,81220,119038,174464,255694,374742,

%U 549215,804918,1179670,1728895,2533823,3713502,5442406,7976239,11689751,17132167

%N G.f. = b(2)*b(4)*b(6)/(x^8+x^6-x^5-x^3-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_2 - 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="/A266333/b266333.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_08">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,1,0,1,-1,0,-1).

%F G.f.: (1+x)^3*(1-x+x^2)*(1+x^2)*(1+x+x^2) / ((1-x)*(1-x-x^3)*(1+x+x^2+x^3+x^4)). - _Colin Barker_, Dec 29 2015

%p gf:= b(2)*b(4)*b(6)/(x^8+x^6-x^5-x^3-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[4] b[6]/(x^8 + x^6 - x^5 - x^3 - 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(4)*b(6)/(x^8+x^6-x^5-x^3-x+1))); // _Bruno Berselli_, Dec 29 2015

%o (PARI) Vec((1+x)^3*(1-x+x^2)*(1+x^2)*(1+x+x^2) / ((1-x)*(1-x-x^3)*(1+x+x^2+x^3+x^4)) + O(x^50)) \\ _Colin Barker_, Dec 29 2015

%Y Cf. similar sequences listed in A265055.

%K nonn,easy

%O 0,2

%A _Alois P. Heinz_, Dec 27 2015