A164069 Number of reduced words of length n in Coxeter group on 36 generators S_i with relations (S_i)^2 = (S_i S_j)^6 = I.

1, 36, 1260, 44100, 1543500, 54022500, 1890786870, 66177518400, 2316212372880, 81067406061600, 2837358267534000, 99307506301920000, 3475761563405646270, 121651614218556733500, 4257805080127526578980, 149023128191211379381500

Offset: 0

Author

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

Keywords

• nonn

Comments

The initial terms coincide with those of A170755, 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..645
• Index entries for linear recurrences with constant coefficients, signature (34,34,34,34,34,-595).

Programs

GAP

a:=[36, 1260, 44100, 1543500, 54022500, 1890786870];; for n in [7..30] do a[n]:=34*(a[n-1] +a[n-2]+a[n-3]+a[n-4]+a[n-5]) -595*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)/(1-35*t+629*t^6-595*t^7) )); // G. C. Greubel, Aug 13 2019

Maple

seq(coeff(series((1+t)*(1-t^6)/(1-35*t+629*t^6-595*t^7), t, n+1), t, n), n = 0 .. 30); # G. C. Greubel, Aug 13 2019

Mathematica

CoefficientList[Series[(1+t)*(1-t^6)/(1-35*t+629*t^6-595*t^7), {t,0,30}], t] (* G. C. Greubel, Sep 09 2017 *)
coxG[{6, 595, -34}] (* The coxG program is at A169452 *) (* G. C. Greubel, Aug 13 2019 *)

PARI

t='t+O('t^50); Vec((1+t)*(1-t^6)/(1-35*t+629*t^6-595*t^7)) \\ G. C. Greubel, Sep 09 2017

Sage

def A164069_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1+t)*(1-t^6)/(1-35*t+629*t^6-595*t^7)).list()
A164069_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)/(595*t^6 - 34*t^5 - 34*t^4 - 34*t^3 - 34*t^2 - 34*t + 1).

a(n) = -595*a(n-6) + 34*Sum_{k=1..5} a(n-k). - Wesley Ivan Hurt, May 11 2021