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Expansion of b(2)*b(6)/(1 - 2*x + x^3 - x^4 - x^5 + 2*x^6), where b(k) = (1-x^k)/(1-x).
2

%I #33 Sep 08 2022 08:46:14

%S 1,4,10,21,41,79,150,282,527,982,1829,3405,6337,11790,21932,40797,

%T 75888,141161,262573,488407,908474,1689830,3143211,5846606,10875117,

%U 20228513,37626513,69988066,130182920,242149745,450416216,837807065,1558382345,2898705007,5391803070

%N Expansion of b(2)*b(6)/(1 - 2*x + x^3 - x^4 - x^5 + 2*x^6), where b(k) = (1-x^k)/(1-x).

%C This is the Poincaré series [or Poincare series] for the quasi-Lannér diagram QL4_20 - 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="/A264079/b264079.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_06">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,-1,1,1,-2).

%F G.f.: (1 + x)^2*(1 - x + x^2)*(1 + x + x^2)/((1 - x)*(1 - x - x^2 - x^4 - 2*x^5)).

%F a(n) = 2*a(n-1) - a(n-3) + a(n-4) + a(n-5) - 2*a(n-6) for n>6.

%t CoefficientList[Series[(1 + x)^2 (1 - x + x^2) (1 + x + x^2)/((1 - x) (1 - x - x^2 - x^4 - 2 x^5)), {x, 0, 40}], x]

%t LinearRecurrence[{2,0,-1,1,1,-2},{1,4,10,21,41,79,150},40] (* _Harvey P. Dale_, Jan 18 2021 *)

%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)/(1-2*x+x^3-x^4-x^5+2*x^6)));

%Y Cf. similar sequences listed in A265055.

%K nonn,easy

%O 0,2

%A _Bruno Berselli_, Dec 28 2015