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A167931 Number of reduced words of length n in Coxeter group on 20 generators S_i with relations (S_i)^2 = (S_i S_j)^16 = I.

%I A167931 #17 Sep 08 2022 08:45:48
%S A167931 1,20,380,7220,137180,2606420,49521980,940917620,17877434780,
%T A167931 339671260820,6453753955580,122621325156020,2329805177964380,
%U A167931 44266298381323220,841059669245141180,15980133715657682420
%N A167931 Number of reduced words of length n in Coxeter group on 20 generators S_i with relations (S_i)^2 = (S_i S_j)^16 = I.
%C A167931 The initial terms coincide with those of A170739, although the two sequences are eventually different.
%C A167931 Computed with MAGMA using commands similar to those used to compute A154638.
%H A167931 G. C. Greubel, <a href="/A167931/b167931.txt">Table of n, a(n) for n = 0..500</a>
%H A167931 <a href="/index/Rec#order_16">Index entries for linear recurrences with constant coefficients</a>, signature (18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, -171).
%F A167931 G.f.: (t^16 + 2*t^15 + 2*t^14 + 2*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)/( 171*t^16 - 18*t^15 - 18*t^14 - 18*t^13 - 18*t^12 - 18*t^11 - 18*t^10 - 18*t^9 - 18*t^8 - 18*t^7 - 18*t^6 - 18*t^5 - 18*t^4 - 18*t^3 - 18*t^2 - 18*t + 1).
%F A167931 G.f.: (1+x)*(1-x^16)/(1 -19*x +189*x^16 -171*x^17). - _G. C. Greubel_, Apr 25 2019
%t A167931 CoefficientList[Series[(1+x)*(1-x^16)/(1-19*x+189*x^16-171*x^17), {x, 0, 20}], x] (* _G. C. Greubel_, Jul 01 2016, modified Apr 25 2019 *)
%t A167931 coxG[{16, 171, -18, 20}] (* The coxG program is at A169452 *) (* _G. C. Greubel_, Apr 25 2019 *)
%o A167931 (PARI) my(x='x+O('x^20)); Vec((1+x)*(1-x^16)/(1-19*x+189*x^16-171*x^17)) \\ _G. C. Greubel_, Apr 25 2019
%o A167931 (Magma) R<x>:=PowerSeriesRing(Integers(), 20); Coefficients(R!( (1+x)*(1-x^16)/(1-19*x+189*x^16-171*x^17) ));  // _G. C. Greubel_, Apr 25 2019
%o A167931 (Sage) ((1+x)*(1-x^16)/(1-19*x+189*x^16-171*x^17)).series(x, 20).coefficients(x, sparse=False) # _G. C. Greubel_, Apr 25 2019
%Y A167931 Cf. A154638, A170739.
%K A167931 nonn
%O A167931 0,2
%A A167931 _John Cannon_ and _N. J. A. Sloane_, Dec 03 2009