A164036 Number of reduced words of length n in Coxeter group on 32 generators S_i with relations (S_i)^2 = (S_i S_j)^6 = I.

1, 32, 992, 30752, 953312, 29552672, 916132336, 28400087040, 880402222080, 27292454123520, 846065620239360, 26228020042137600, 813068181562751760, 25205099996403756000, 781357677295456980000, 24222074895781408504800, 750883915657732812602400

Offset: 0

Author

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

Keywords

• nonn

Comments

The initial terms coincide with those of A170751, 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..665
• Index entries for linear recurrences with constant coefficients, signature (30,30,30,30,30,-465).

Programs

GAP

a:=[32, 992, 30752, 953312, 29552672, 916132336];; for n in [7..30] do a[n]:=30*(a[n-1] +a[n-2]+a[n-3]+a[n-4]+a[n-5]) -465*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)/(1-31*t+495*t^6-465*t^7) )); // G. C. Greubel, Aug 13 2019

Maple

seq(coeff(series((1+t)*(1-t^6)/(1-31*t+495*t^6-465*t^7), t, n+1), t, n), n = 0 .. 30); # G. C. Greubel, Aug 13 2019

Mathematica

CoefficientList[Series[(1+t)*(1-t^6)/(1-31*t+495*t^6-465*t^7), {t,0,30}], t] (* G. C. Greubel, Sep 08 2017 *)
coxG[{6, 465, -30}] (* The coxG program is at A169452 *) (* G. C. Greubel, Aug 13 2019 *)

PARI

my(t='t+O('t^30)); Vec((1+t)*(1-t^6)/(1-31*t+495*t^6-465*t^7)) \\ G. C. Greubel, Sep 08 2017

Sage

def A164036_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1+t)*(1-t^6)/(1-31*t+495*t^6-465*t^7)).list()
A164036_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)/(465*t^6 - 30*t^5 - 30*t^4 - 30*t^3 - 30*t^2 - 30*t + 1).

a(n) = -465*a(n-6) + 30*Sum_{k=1..5} a(n-k). - Wesley Ivan Hurt, May 11 2021