A163923 Number of reduced words of length n in Coxeter group on 7 generators S_i with relations (S_i)^2 = (S_i S_j)^6 = I.

1, 7, 42, 252, 1512, 9072, 54411, 326340, 1957305, 11739420, 70410060, 422301600, 2532857460, 15191434125, 91114353750, 546480693675, 3277652052150, 19658522431800, 117906811965600, 707175035973000, 4241455800274875

Offset: 0

Author

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

Keywords

• nonn

Comments

The initial terms coincide with those of A003949, 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..1000
• Index entries for linear recurrences with constant coefficients, signature (5,5,5,5,5,-15).

Programs

GAP

a:=[7,42,252,1512,9072,54411];; for n in [7..30] do a[n]:=5*(a[n-1] +a[n-2]+a[n-3]+a[n-4]+a[n-5]) -15*a[n-6]; od; Concatenation([1], a); # G. C. Greubel, Aug 10 2019

Magma

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+t)*(1-t^6)/(1-6*t+20*t^6-15*t^7) )); // G. C. Greubel, Aug 10 2019

Maple

seq(coeff(series((1+t)*(1-t^6)/(1-6*t+20*t^6-15*t^7), t, n+1), t, n), n = 0 .. 30); # G. C. Greubel, Aug 10 2019

Mathematica

coxG[{6,15,-5}] (* The coxG program is at A169452 *) (* Harvey P. Dale, Aug 18 2015 *)
CoefficientList[Series[(1+t)*(1-t^6)/(1-6*t+20*t^6-15*t^7), {t,0,30}], t] (* G. C. Greubel, Aug 08 2017 *)

PARI

my(t='t+O('t^30)); Vec((1+t)*(1-t^6)/(1-6*t+20*t^6-15*t^7)) \\ G. C. Greubel, Aug 08 2017

Sage

def A163923_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1+t)*(1-t^6)/(1-6*t+20*t^6-15*t^7)).list()
A163923_list(30) # G. C. Greubel, Aug 10 2019

Formula

G.f.: (t^6 + 2*t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(15*t^6 - 5*t^5 - 5*t^4 - 5*t^3 - 5*t^2 - 5*t + 1).

a(n) = 5*a(n-1)+5*a(n-2)+5*a(n-3)+5*a(n-4)+5*a(n-5)-15*a(n-6). - Wesley Ivan Hurt, Apr 23 2021