A163977 Number of reduced words of length n in Coxeter group on 21 generators S_i with relations (S_i)^2 = (S_i S_j)^6 = I.

1, 21, 420, 8400, 168000, 3360000, 67199790, 1343991600, 26879748210, 537593288400, 10751832252000, 215035974720000, 4300706088043890, 86013853634593500, 1720271710182898110, 34405326953812846500

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 (19,19,19,19,19,-190).

Programs

GAP

a:=[21, 420, 8400, 168000, 3360000, 67199790];; for n in [7..30] do a[n]:=19*(a[n-1] +a[n-2]+a[n-3]+a[n-4]+a[n-5]) -190*a[n-6]; od; Concatenation([1], a); # G. C. Greubel, Aug 11 2019

Magma

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+t)*(1-t^6)/(1-20*t+209*t^6-190*t^7) )); // G. C. Greubel, Aug 11 2019

Maple

seq(coeff(series((1+t)*(1-t^6)/(1-20*t+209*t^6-190*t^7), t, n+1), t, n), n = 0 .. 30); # G. C. Greubel, Aug 11 2019

Mathematica

CoefficientList[Series[(1+t)*(1-t^6)/(1-20*t+209*t^6-190*t^7), {t,0,30}], t] (* G. C. Greubel, Aug 24 2017 *)
coxG[{6, 190, -19}] (* The coxG program is at A169452 *) (* G. C. Greubel, Aug 11 2019 *)

PARI

my(t='t+O('t^30)); Vec((1+t)*(1-t^6)/(1-20*t+209*t^6-190*t^7)) \\ G. C. Greubel, Aug 24 2017

Sage

def A163977_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1+t)*(1-t^6)/(1-20*t+209*t^6-190*t^7)).list()
A163977_list(30) # G. C. Greubel, Aug 11 2019

Formula

G.f.: (t^6 + 2*t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(190*t^6 - 19*t^5 - 19*t^4 - 19*t^3 - 19*t^2 - 19*t + 1).

a(n) = -190*a(n-6) + 19*Sum_{k=1..5} a(n-k). - Wesley Ivan Hurt, May 11 2021