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G.f. = b(2)^2*b(6)/(x^7+x^6-x^5-x^2-x+1), where b(k) = (1-x^k)/(1-x).
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%I #23 Sep 08 2022 08:46:15

%S 1,4,9,17,30,52,88,145,237,386,628,1020,1653,2677,4334,7016,11356,

%T 18377,29737,48118,77860,125984,203849,329837,533690,863532,1397228,

%U 2260765,3657997,5918766,9576768,15495540,25072313,40567857,65640174,106208036,171848216

%N G.f. = b(2)^2*b(6)/(x^7+x^6-x^5-x^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_4 - 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="/A266335/b266335.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_07">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,0,0,1,-1,-1).

%p gf:= b(2)^2*b(6)/(x^7+x^6-x^5-x^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]^2 b[6]/(x^7 + x^6 - x^5 - x^2 - x + 1), {x, 0, 40}], x] (* _Bruno Berselli_, Dec 29 2015 *)

%t LinearRecurrence[{1,1,0,0,1,-1,-1},{1,4,9,17,30,52,88,145},40] (* _Harvey P. Dale_, Mar 23 2020 *)

%o (Magma) /* By definition: */ m:=40; R<x>:=PowerSeriesRing(Integers(), m); b:=func<k|(1-x^k)/(1-x)>; Coefficients(R!(b(2)^2*b(6)/(x^7+x^6-x^5-x^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