A266353 - OEIS

%I #21 Sep 08 2022 08:46:15

%S 1,4,10,20,35,57,89,136,205,306,454,671,989,1455,2138,3139,4606,6756,

%T 9907,14525,21293,31212,45749,67054,98278,144039,211105,309395,453446,

%U 664563,973970,1427428,2092003,3065985,4493425,6585440,9651437,14144874,20730326,30381775

%N Expansion of b(3)*b(4)/(1 - 2*x + x^2 - x^3 + x^4), where b(k) = (1-x^k)/(1-x).

%C This is the Poincaré series [or Poincare series] for the quasi-Lannér diagram QL4_17 - see Table 7.8 in Maxim Chapovalov, Dimitry Leites and Rafael Stekolshchik (2009), or equivalently Table 6 in the shorter version, Maxim Chapovalov, Dimitry Leites and Rafael Stekolshchik (2010).

%H Colin Barker, <a href="/A266353/b266353.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 [or Poincare series] of the hyperbolic Coxeter groups with finite volume of fundamental domains</a>, arXiv:0906.1596 [math.RT], 2009, page 31.

%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, Volume 17, Supplement 1 (2010), page 186.

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,1,-1).

%F G.f.: (1 + x)*(1 + x^2)*(1 + x + x^2)/((1 - x)*(1 - x - x^3)).

%F a(n) = 2*a(n-1) - a(n-2) + a(n-3) - a(n-4) for n>5.

%F a(n) = a(n-1) + a(n-3) + 12 for n>4. - _Greg Dresden_, Feb 09 2020

%t CoefficientList[Series[(1 + x) (1 + x^2) (1 + x + x^2)/((1 - x) (1 - x - x^3)), {x, 0, 40}], x]

%o (Magma) /* By definition: */ m:=40; R<x>:=PowerSeriesRing(Integers(), m); b:=func<k|(1-x^k)/(1-x)>; Coefficients(R!(b(3)*b(4)/(1-2*x+x^2-x^3+x^4)));

%Y Cf. similar sequences listed in A265055.

%K nonn,easy

%O 0,2

%A _Bruno Berselli_, Dec 28 2015