A164277 Number of reduced words of length n in Coxeter group on 44 generators S_i with relations (S_i)^2 = (S_i S_j)^6 = I.

1, 44, 1892, 81356, 3498308, 150427244, 6468370546, 278139892800, 11960013642192, 514280511441312, 22114058759539824, 950904387665438976, 40888882692839511330, 1758221698790838894228, 75603521996953503778764

Offset: 0

Author

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

Keywords

• nonn

Comments

The initial terms coincide with those of A170763, 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..610
• Index entries for linear recurrences with constant coefficients, signature (42,42,42,42,42,-903).

Programs

GAP

a:=[44, 1892, 81356, 3498308, 150427244, 6468370546];; for n in [7..30] do a[n]:=42*(a[n-1] +a[n-2]+a[n-3]+a[n-4]+a[n-5]) -903*a[n-6]; od; Concatenation([1], a); # G. C. Greubel, Aug 16 2019

Magma

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+t)*(1-t^6)/(1-43*t+945*t^6-903*t^7) )); // G. C. Greubel, Aug 16 2019

Maple

seq(coeff(series((1+t)*(1-t^6)/(1-43*t+945*t^6-903*t^7), t, n+1), t, n), n = 0 .. 30); # G. C. Greubel, Aug 16 2019

Mathematica

CoefficientList[Series[(1+t)*(1-t^6)/(1-43*t+945*t^6-903*t^7), {t,0,30}], t] (* G. C. Greubel, Sep 12 2017 *)
coxG[{6, 903, -42}] (* The coxG program is at A169452 *) (* G. C. Greubel, Aug 16 2019 *)

PARI

my(t='t+O('t^30)); Vec((1+t)*(1-t^6)/(1-43*t+945*t^6-903*t^7)) \\ G. C. Greubel, Sep 12 2017

Sage

def A164277_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1+t)*(1-t^6)/(1-43*t+945*t^6-903*t^7)).list()
A164277_list(30) # G. C. Greubel, Aug 16 2019

Formula

G.f.: (t^6 + 2*t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(903*t^6 - 42*t^5 - 42*t^4 - 42*t^3 - 42*t^2 - 42*t + 1).

a(n) = -903*a(n-6) + 42*Sum_{k=1..5} a(n-k). - Wesley Ivan Hurt, May 11 2021