A163965 Number of reduced words of length n in Coxeter group on 17 generators S_i with relations (S_i)^2 = (S_i S_j)^6 = I.

1, 17, 272, 4352, 69632, 1114112, 17825656, 285208320, 4563298440, 73012220160, 1168186644480, 18690844262400, 299051235428280, 4784783402808000, 76555952624637000, 1224885932940283200, 19598025983313945600

Offset: 0

Author

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

Keywords

• nonn

Comments

The initial terms coincide with those of A170736, 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..825
• Index entries for linear recurrences with constant coefficients, signature (15,15,15,15,15,-120).

Programs

GAP

a:=[17, 272, 4352, 69632, 1114112, 17825656];; for n in [7..30] do a[n]:=15*(a[n-1] +a[n-2]+a[n-3]+a[n-4]+a[n-5]) -120*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)/(1-16*t+135*t^6-120*t^7) )); // G. C. Greubel, Aug 11 2019

Maple

seq(coeff(series((1+t)*(1-t^6)/(1-16*t+135*t^6-120*t^7), t, n+1), t, n), n = 0 .. 30); # G. C. Greubel, Aug 11 2019

Mathematica

CoefficientList[Series[(1+t)*(1-t^6)/(1-16*t+135*t^6-120*t^7), {t,0,30}], t] (* G. C. Greubel, Aug 23 2017 *)
coxG[{6, 120, -15}] (* 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)/(1-16*t+135*t^6-120*t^7)) \\ G. C. Greubel, Aug 23 2017

Sage

def A163965_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1+t)*(1-t^6)/(1-16*t+135*t^6-120*t^7)).list()
A163965_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)/(120*t^6 - 15*t^5 - 15*t^4 - 15*t^3 - 15*t^2 - 15*t + 1).

a(n) = -120*a(n-6) + 15*Sum_{k=1..5} a(n-k). - Wesley Ivan Hurt, May 11 2021