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A166691 Number of reduced words of length n in Coxeter group on 39 generators S_i with relations (S_i)^2 = (S_i S_j)^12 = I.

%I A166691 #20 Sep 08 2022 08:45:48
%S A166691 1,39,1482,56316,2140008,81320304,3090171552,117426518976,
%T A166691 4462207721088,169563893401344,6443427949251072,244850262071540736,
%U A166691 9304309958718547227,353563778431304766468,13435423580389580056521
%N A166691 Number of reduced words of length n in Coxeter group on 39 generators S_i with relations (S_i)^2 = (S_i S_j)^12 = I.
%C A166691 The initial terms coincide with those of A170758, although the two sequences are eventually different.
%C A166691 Computed with MAGMA using commands similar to those used to compute A154638.
%H A166691 G. C. Greubel, <a href="/A166691/b166691.txt">Table of n, a(n) for n = 0..500</a>
%H A166691 <a href="/index/Rec#order_12">Index entries for linear recurrences with constant coefficients</a>, signature (37, 37, 37, 37, 37, 37, 37, 37, 37, 37, 37, -703).
%F A166691 G.f.: (t^12 + 2*t^11 + 2*t^10 + 2*t^9 + 2*t^8 + 2*t^7 + 2*t^6 + 2*t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(703*t^12 - 37*t^11 - 37*t^10 - 37*t^9 -37*t^8 -37*t^7 -37*t^6 - 37*t^5 - 37*t^4 - 37*t^3 - 37*t^2 - 37*t + 1).
%F A166691 G.f.: (1+x)*(1-x^12)/(1 -38*x +740*x^12 -703*x^13). - _G. C. Greubel_, Apr 26 2019
%F A166691 a(n) = -703*a(n-12) + 37*Sum_{k=1..11} a(n-k). - _Wesley Ivan Hurt_, May 06 2021
%t A166691 CoefficientList[Series[(1+x)*(1-x^12)/(1-38*x+740*x^12-703*x^13), {x, 0, 20}], x] (* _G. C. Greubel_, May 23 2016, modified Apr 26 2019 *)
%t A166691 coxG[{12,703,-37}] (* The coxG program is at A169452 *) (* _Harvey P. Dale_, Jan 10 2017 *)
%o A166691 (PARI) my(x='x+O('x^20)); Vec((1+x)*(1-x^12)/(1-38*x+740*x^12-703*x^13)) \\ _G. C. Greubel_, Apr 26 2019
%o A166691 (Magma) R<x>:=PowerSeriesRing(Integers(), 20); Coefficients(R!( (1+x)*(1-x^12)/(1-38*x+740*x^12-703*x^13) )); // _G. C. Greubel_, Apr 26 2019
%o A166691 (Sage) ((1+x)*(1-x^12)/(1-38*x+740*x^12-703*x^13)).series(x, 20).coefficients(x, sparse=False) # _G. C. Greubel_, Apr 26 2019
%K A166691 nonn
%O A166691 0,2
%A A166691 _John Cannon_ and _N. J. A. Sloane_, Dec 03 2009