A164017 Number of reduced words of length n in Coxeter group on 27 generators S_i with relations (S_i)^2 = (S_i S_j)^6 = I.

1, 27, 702, 18252, 474552, 12338352, 320796801, 8340707700, 216858163275, 5638306085100, 146595798051300, 3811486585140000, 99098542944724050, 2576559301574090625, 66990468651299212500, 1741750282005552804375

Offset: 0

Author

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

Keywords

• nonn

Comments

The initial terms coincide with those of A170746, 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..705
• Index entries for linear recurrences with constant coefficients, signature (25,25,25,25,25,-325).

Programs

GAP

a:=[27, 702, 18252, 474552, 12338352, 320796801];; for n in [7..30] do a[n]:=25*(a[n-1] +a[n-2]+a[n-3]+a[n-4]+a[n-5]) -325*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)/(1-26*t+350*t^6-325*t^7) )); // G. C. Greubel, Aug 13 2019

Maple

seq(coeff(series((1+t)*(1-t^6)/(1-26*t+350*t^6-325*t^7), t, n+1), t, n), n = 0 .. 30); # G. C. Greubel, Aug 13 2019

Mathematica

coxG[{6,325,-25}] (* The coxG program is at A169452 *) (* Harvey P. Dale, Jun 26 2017 *)
CoefficientList[Series[(1+t)*(1-t^6)/(1-26*t+350*t^6-325*t^7), {t,0,30}], t] (* G. C. Greubel, Sep 07 2017 *)

PARI

my(t='t+O('t^30)); Vec((1+t)*(1-t^6)/(1-26*t+350*t^6-325*t^7)) \\ G. C. Greubel, Sep 07 2017

Sage

def A164017_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1+t)*(1-t^6)/(1-26*t+350*t^6-325*t^7)).list()
A164017_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)/(325*t^6 - 25*t^5 - 25*t^4 - 25*t^3 - 25*t^2 - 25*t + 1).

a(n) = -325*a(n-6) + 25*Sum_{k=1..5} a(n-k). - Wesley Ivan Hurt, May 11 2021