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A167048 Number of reduced words of length n in Coxeter group on 18 generators S_i with relations (S_i)^2 = (S_i S_j)^13 = I.

%I A167048 #19 Sep 08 2022 08:45:48
%S A167048 1,18,306,5202,88434,1503378,25557426,434476242,7386096114,
%T A167048 125563633938,2134581776946,36287890208082,616894133537394,
%U A167048 10487200270135545,178282404592301664,3030800878069084224,51523614927173682720
%N A167048 Number of reduced words of length n in Coxeter group on 18 generators S_i with relations (S_i)^2 = (S_i S_j)^13 = I.
%C A167048 The initial terms coincide with those of A170737, although the two sequences are eventually different.
%C A167048 Computed with MAGMA using commands similar to those used to compute A154638.
%H A167048 G. C. Greubel, <a href="/A167048/b167048.txt">Table of n, a(n) for n = 0..500</a>
%H A167048 <a href="/index/Rec#order_13">Index entries for linear recurrences with constant coefficients</a>, signature (16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, -136).
%F A167048 G.f.: (t^13 + 2*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)/(136*t^13 - 16*t^12 - 16*t^11 - 16*t^10 - 16*t^9 - 16*t^8 - 16*t^7 - 16*t^6 - 16*t^5 - 16*t^4 - 16*t^3 - 16*t^2 - 16*t + 1).
%F A167048 G.f.: (1+x)*(1-x^13)/(1 - 17*x + 152*x^13 - 136*x^14). - _G. C. Greubel_, Apr 26 2019
%F A167048 a(n) = -136*a(n-13) + 16*Sum_{k=1..12} a(n-k). - _Wesley Ivan Hurt_, May 06 2021
%t A167048 CoefficientList[Series[(1+x)*(1-x^13)/(1-17*x+152*x^13-136*x^14), {x, 0, 20}], x] (* _G. C. Greubel_, May 30 2016, modified Apr 26 2019 *)
%t A167048 coxG[{13, 136, -16}] (* The coxG program is at A169452 *) (* _G. C. Greubel_, Apr 26 2019 *)
%o A167048 (PARI) my(x='x+O('x^20)); Vec((1+x)*(1-x^13)/(1-17*x+152*x^13-136*x^14)) \\ _G. C. Greubel_, Apr 26 2019
%o A167048 (Magma) R<x>:=PowerSeriesRing(Integers(), 20); Coefficients(R!( (1+x)*(1-x^13)/(1-17*x+152*x^13-136*x^14) )); // _G. C. Greubel_, Apr 26 2019
%o A167048 (Sage) ((1+x)*(1-x^13)/(1-17*x+152*x^13-136*x^14)).series(x, 20).coefficients(x, sparse=False) # _G. C. Greubel_, Apr 26 2019
%K A167048 nonn
%O A167048 0,2
%A A167048 _John Cannon_ and _N. J. A. Sloane_, Dec 03 2009