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A167929 Number of reduced words of length n in Coxeter group on 19 generators S_i with relations (S_i)^2 = (S_i S_j)^16 = I.

%I A167929 #21 Sep 08 2022 08:45:48
%S A167929 1,19,342,6156,110808,1994544,35901792,646232256,11632180608,
%T A167929 209379250944,3768826516992,67838877305856,1221099791505408,
%U A167929 21979796247097344,395636332447752192,7121453984059539456
%N A167929 Number of reduced words of length n in Coxeter group on 19 generators S_i with relations (S_i)^2 = (S_i S_j)^16 = I.
%C A167929 The initial terms coincide with those of A170738, although the two sequences are eventually different.
%C A167929 Computed with MAGMA using commands similar to those used to compute A154638.
%H A167929 G. C. Greubel, <a href="/A167929/b167929.txt">Table of n, a(n) for n = 0..500</a>
%H A167929 <a href="/index/Rec#order_16">Index entries for linear recurrences with constant coefficients</a>, signature (17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, -153).
%F A167929 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)/( 153*t^16 - 17*t^15 - 17*t^14 - 17*t^13 - 17*t^12 - 17*t^11 - 17*t^10 - 17*t^9 - 17*t^8 - 17*t^7 - 17*t^6 - 17*t^5 - 17*t^4 - 17*t^3 - 17*t^2 - 17*t + 1).
%F A167929 G.f.: (1+x)*(1-x^16)/(1 - 18*x + 170*x^16 - 153*x^17). - _G. C. Greubel_, Apr 26 2019
%F A167929 a(n) = -153*a(n-16) + 17*Sum_{k=1..15} a(n-k). - _Wesley Ivan Hurt_, May 06 2021
%t A167929 CoefficientList[Series[(1+x)*(1-x^16)/(1-18*x+170*x^16-153*x^17), {x, 0, 20}], x] (* _G. C. Greubel_, Jul 01 2016, modified Apr 26 2019 *)
%t A167929 coxG[{16, 153, -17}] (* The coxG program is at A169452 *) (* _G. C. Greubel_, Apr 26 2019 *)
%o A167929 (PARI) my(x='x+O('x^20)); Vec((1+x)*(1-x^16)/(1-18*x+170*x^16-153*x^17)) \\ _G. C. Greubel_, Apr 26 2019
%o A167929 (Magma) R<x>:=PowerSeriesRing(Integers(), 20); Coefficients(R!( (1+x)*(1-x^16)/(1-18*x+170*x^16-153*x^17) )); // _G. C. Greubel_, Apr 26 2019
%o A167929 (Sage) ((1+x)*(1-x^16)/(1-18*x+170*x^16-153*x^17)).series(x, 20).coefficients(x, sparse=False) # _G. C. Greubel_, Apr 26 2019
%Y A167929 Cf. A154638, A170738.
%K A167929 nonn
%O A167929 0,2
%A A167929 _John Cannon_ and _N. J. A. Sloane_, Dec 03 2009