A167933 - OEIS

%I #17 Sep 08 2022 08:45:48

%S 1,21,420,8400,168000,3360000,67200000,1344000000,26880000000,

%T 537600000000,10752000000000,215040000000000,4300800000000000,

%U 86016000000000000,1720320000000000000,34406400000000000000

%N Number of reduced words of length n in Coxeter group on 21 generators S_i with relations (S_i)^2 = (S_i S_j)^16 = I.

%C The initial terms coincide with those of A170740, although the two sequences are eventually different.

%C Computed with MAGMA using commands similar to those used to compute A154638.

%H G. C. Greubel, <a href="/A167933/b167933.txt">Table of n, a(n) for n = 0..500</a>

%H <a href="/index/Rec#order_16">Index entries for linear recurrences with constant coefficients</a>, signature (19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, -190).

%F 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)/( 190*t^16 - 19*t^15 - 19*t^14 - 19*t^13 - 19*t^12 - 19*t^11 - 19*t^10 - 19*t^9 - 19*t^8 - 19*t^7 - 19*t^6 - 19*t^5 - 19*t^4 - 19*t^3 - 19*t^2 - 19*t + 1).

%F G.f.: (1+x)*(1-x^16)/(1 -20*x +209*x^16 -190*x^17). - _G. C. Greubel_, Apr 25 2019

%t CoefficientList[Series[(1+x)*(1-x^16)/(1-20*x+209*x^16-190*x^17), {x, 0, 20}], x] (* _G. C. Greubel_, Jul 01 2016, modified Apr 25 2019 *)

%t coxG[{16, 190, -19, 20}] (* The coxG program is at A169452 *) (* _G. C. Greubel_, Apr 25 2019 *)

%o (PARI) my(x='x+O('x^20)); Vec((1+x)*(1-x^16)/(1-20*x+209*x^16-190*x^17)) \\ _G. C. Greubel_, Apr 25 2019

%o (Magma) R<x>:=PowerSeriesRing(Integers(), 20); Coefficients(R!( (1+x)*(1-x^16)/(1-20*x+209*x^16-190*x^17) )); // _G. C. Greubel_, Apr 25 2019

%o (Sage) ((1+x)*(1-x^16)/(1-20*x+209*x^16-190*x^17)).series(x, 20).coefficients(x, sparse=False) # _G. C. Greubel_, Apr 25 2019

%Y Cf. A154638, A170740.

%K nonn

%O 0,2

%A _John Cannon_ and _N. J. A. Sloane_, Dec 03 2009