A164112 Number of reduced words of length n in Coxeter group on 42 generators S_i with relations (S_i)^2 = (S_i S_j)^6 = I.

1, 42, 1722, 70602, 2894682, 118681962, 4865959581, 199504307520, 8179675161840, 335366622329760, 13750029083987280, 563751092750630400, 23113790715369815580, 947665251746544828000, 38854268450681230932000, 1593024724769968897327200, 65314002165544342871757600

Offset: 0

Author

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

Keywords

• nonn

Comments

The initial terms coincide with those of A170761, 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..615
• Index entries for linear recurrences with constant coefficients, signature (40,40,40,40,40,-820).

Programs

GAP

a:=[42, 1722, 70602, 2894682, 118681962, 4865959581];; for n in [7..30] do a[n]:=40*(a[n-1] +a[n-2]+a[n-3]+a[n-4]+a[n-5]) -820*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)/(1-41*t+860*t^6-820*t^7) )); // G. C. Greubel, Aug 16 2019

Maple

seq(coeff(series((1+t)*(1-t^6)/(1-41*t+860*t^6-820*t^7), t, n+1), t, n), n = 0 .. 30); # G. C. Greubel, Aug 16 2019

Mathematica

CoefficientList[Series[(1+t)*(1-t^6)/(1-41*t+860*t^6-820*t^7), {t,0,30}], t] (* G. C. Greubel, Sep 11 2017 *)
coxG[{6,820,-40}] (* The coxG program is at A169452 *) (* Harvey P. Dale, Jan 16 2018 *)

PARI

my(t='t+O('t^30)); Vec((1+t)*(1-t^6)/(1-41*t+860*t^6-820*t^7)) \\ G. C. Greubel, Sep 11 2017

Sage

def A164112_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1+t)*(1-t^6)/(1-41*t+860*t^6-820*t^7)).list()
A164112_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)/(820*t^6 - 40*t^5 - 40*t^4 - 40*t^3 - 40*t^2 - 40*t + 1).

a(n) = -820*a(n-6) + 40*Sum_{k=1..5} a(n-k). - Wesley Ivan Hurt, May 11 2021