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A163645 Number of reduced words of length n in Coxeter group on 37 generators S_i with relations (S_i)^2 = (S_i S_j)^5 = I.

%I A163645 #19 Sep 08 2022 08:45:46
%S A163645 1,37,1332,47952,1726272,62145126,2237200560,80538357690,
%T A163645 2899349827920,104375476044000,3757476898626570,135267719763613500,
%U A163645 4869585762918574950,175303210136598476100,6310847981816367469200
%N A163645 Number of reduced words of length n in Coxeter group on 37 generators S_i with relations (S_i)^2 = (S_i S_j)^5 = I.
%C A163645 The initial terms coincide with those of A170756, although the two sequences are eventually different.
%C A163645 Computed with MAGMA using commands similar to those used to compute A154638.
%H A163645 G. C. Greubel, <a href="/A163645/b163645.txt">Table of n, a(n) for n = 0..640</a>
%H A163645 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (35, 35, 35, 35, -630).
%F A163645 G.f.: (t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(630*t^5 - 35*t^4 - 35*t^3 - 35*t^2 - 35*t + 1).
%F A163645 a(n) = -630*a(n-5) + 35*Sum_{k=1..4} a(n-k). - _Wesley Ivan Hurt_, May 05 2021
%t A163645 CoefficientList[Series[(1+x)*(1-x^5)/(1-36*x+665*x^5-630*x^6), {x,0,20}], x] (* _G. C. Greubel_, Aug 01 2017 *)
%t A163645 coxG[{5,630,-35}] (* The coxG program is at A169452 *) (* _Harvey P. Dale_, Aug 31 2018 *)
%o A163645 (PARI) my(x='x+O('x^20)); Vec((1+x)*(1-x^5)/(1-36*x+665*x^5-630*x^6)) \\ _G. C. Greubel_, Aug 01 2017
%o A163645 (Magma) R<x>:=PowerSeriesRing(Integers(), 20); Coefficients(R!( (1+x)*(1-x^5)/(1-36*x+665*x^5-630*x^6) )); // _G. C. Greubel_, May 22 2019
%o A163645 (Sage) ((1+x)*(1-x^5)/(1-36*x+665*x^5-630*x^6)).series(x, 20).coefficients(x, sparse=False) # _G. C. Greubel_, May 22 2019
%o A163645 (GAP) a:=[37, 1332, 47952, 1726272, 62145126];; for n in [6..20] do a[n]:=18*(a[n-1]+a[n-2] +a[n-3]+a[n-4] -18*a[n-5]); od; Concatenation([1], a); # _G. C. Greubel_, May 22 2019
%K A163645 nonn
%O A163645 0,2
%A A163645 _John Cannon_ and _N. J. A. Sloane_, Dec 03 2009