A163959 Number of reduced words of length n in Coxeter group on 14 generators S_i with relations (S_i)^2 = (S_i S_j)^6 = I.

1, 14, 182, 2366, 30758, 399854, 5198011, 67572960, 878433192, 11419432752, 148450042104, 1929816959616, 25087183842630, 326127713832648, 4239586491528024, 55113665158705608, 716465177275138200

Offset: 0

Author

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

Keywords

• nonn

Comments

The initial terms coincide with those of A170733, 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..895
• Index entries for linear recurrences with constant coefficients, signature (12,12,12,12,12,-78).

Programs

GAP

a:=[14, 182, 2366, 30758, 399854, 5198011];; for n in [7..30] do a[n]:=12*(a[n-1] +a[n-2]+a[n-3]+a[n-4]+a[n-5]) -78*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)/(1-13*t+90*t^6-78*t^7) )); // G. C. Greubel, Aug 11 2019

Maple

seq(coeff(series((1+t)*(1-t^6)/(1-13*t+90*t^6-78*t^7), t, n+1), t, n), n = 0 .. 30); # G. C. Greubel, Aug 11 2019

Mathematica

CoefficientList[Series[(1+t)*(1-t^6)/(1-13*t+90*t^6-78*t^7), {t,0,30}], t] (* G. C. Greubel, Aug 13 2017 *)
coxG[{6, 78, -12}] (* 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)/(1-13*t+90*t^6-78*t^7)) \\ G. C. Greubel, Aug 13 2017

Sage

def A163959_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1+t)*(1-t^6)/(1-13*t+90*t^6-78*t^7)).list()
A163959_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)/(78*t^6 - 12*t^5 - 12*t^4 - 12*t^3 - 12*t^2 - 12*t + 1).

a(n) = -78*a(n-6) + 12*Sum_{k=1..5} a(n-k). - Wesley Ivan Hurt, May 11 2021