A163503 Number of reduced words of length n in Coxeter group on 21 generators S_i with relations (S_i)^2 = (S_i S_j)^5 = I.

1, 21, 420, 8400, 168000, 3359790, 67191600, 1343748210, 26873288400, 537432252000, 10747974763890, 214946090593500, 4298653734898110, 85967713492846500, 1719247052441058000, 34382796834223386990

Offset: 0

Author

John Cannon and N. J. A. Sloane, Dec 03 2009

Keywords

• nonn

Comments

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

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

Links

• G. C. Greubel, Table of n, a(n) for n = 0..765
• Index entries for linear recurrences with constant coefficients, signature (19, 19, 19, 19, -190).

Programs

Magma

R:=PowerSeriesRing(Integers(), 20); Coefficients(R!( (1+x)*(1-x^5)/(1-20*x+209*x^5-190*x^6) )); // G. C. Greubel, May 16 2019

Mathematica

coxG[{5,190,-19}] (* The coxG program is at A169452 *) (* Harvey P. Dale, Jul 09 2015 *)
CoefficientList[Series[(1+x)*(1-x^5)/(1-20*x+209*x^5-190*x^6), {x,0,20}], x] (* G. C. Greubel, Jul 26 2017 *)

PARI

my(x='x+O('x^20)); Vec((1+x)*(1-x^5)/(1-20*x+209*x^5-190*x^6)) \\ G. C. Greubel, Jul 26 2017

Sage

((1+x)*(1-x^5)/(1-20*x+209*x^5-190*x^6)).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, May 16 2019

Formula

G.f.: (t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(190*t^5 - 19*t^4 - 19*t^3 - 19*t^2 - 19*t + 1).

a(n) = 19*a(n-1)+19*a(n-2)+19*a(n-3)+19*a(n-4)-190*a(n-5). - Wesley Ivan Hurt, May 10 2021