A163995 Number of reduced words of length n in Coxeter group on 26 generators S_i with relations (S_i)^2 = (S_i S_j)^6 = I.

1, 26, 650, 16250, 406250, 10156250, 253905925, 6347640000, 158690797200, 3967264860000, 99181494750000, 2479534200000000, 61988275781355300, 1549704914070300000, 38742573340231207200, 968563095719204700000, 24214046448355276500000, 605350387594249537500000

Offset: 0

Author

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

Keywords

• nonn

Comments

The initial terms coincide with those of A170745, 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..710
• Index entries for linear recurrences with constant coefficients, signature (24,24,24,24,24,-300).

Programs

GAP

a:=[26, 650, 16250, 406250, 10156250, 253905925];; for n in [7..30] do a[n]:=24*(a[n-1] +a[n-2]+a[n-3]+a[n-4]+a[n-5]) -300*a[n-6]; od; Concatenation([1], a); # G. C. Greubel, Aug 13 2019

Magma

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+t)*(1-t^6)/(1-25*t+324*t^6-300*t^7) )); // G. C. Greubel, Aug 13 2019

Maple

seq(coeff(series((1+t)*(1-t^6)/(1-25*t+324*t^6-300*t^7), t, n+1), t, n), n = 0 .. 30); # G. C. Greubel, Aug 13 2019

Mathematica

CoefficientList[Series[(1+t)*(1-t^6)/(1-25*t+324*t^6-300*t^7), {t,0,30}], t] (* G. C. Greubel, Aug 24 2017 *)
coxG[{6, 300, -24}] (* The coxG program is at A169452 *) (* G. C. Greubel, Aug 13 2019 *)

PARI

my(t='t+O('t^30)); Vec((1+t)*(1-t^6)/(1-25*t+324*t^6-300*t^7)) \\ G. C. Greubel, Aug 24 2017

Sage

def A163995_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1+t)*(1-t^6)/(1-25*t+324*t^6-300*t^7)).list()
A163995_list(30) # G. C. Greubel, Aug 13 2019

Formula

G.f.: (t^6 + 2*t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(300*t^6 - 24*t^5 - 24*t^4 - 24*t^3 - 24*t^2 - 24*t + 1).

a(n) = -300*a(n-6) + 24*Sum_{k=1..5} a(n-k). - Wesley Ivan Hurt, May 11 2021