A164350 Number of reduced words of length n in Coxeter group on 49 generators S_i with relations (S_i)^2 = (S_i S_j)^6 = I.

1, 49, 2352, 112896, 5419008, 260112384, 12485393256, 599298819840, 28766340643992, 1380784220911872, 66277636363782144, 3181326245942132736, 152703645428292064680, 7329774290465429385408, 351829132817899422588504, 16887796785286144959221568

Offset: 0

Author

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

Keywords

• nonn

Comments

The initial terms coincide with those of A170768, 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..590
• Index entries for linear recurrences with constant coefficients, signature (47, 47, 47, 47, 47, -1128).

Programs

GAP

a:=[49, 2352, 112896, 5419008, 260112384, 12485393256];; for n in [7..20] do a[n]:=47*(a[n-1] +a[n-2]+a[n-3]+a[n-4]+a[n-5]) -1128*a[n-6]; od; Concatenation([1], a); # G. C. Greubel, Aug 24 2019

Magma

R:=PowerSeriesRing(Integers(), 20); Coefficients(R!( (1+t)*(1-t^6)/(1-48*t+1175*t^6-1128*t^7) )); // G. C. Greubel, Aug 24 2019

Maple

seq(coeff(series((1+t)*(1-t^6)/(1-48*t+1175*t^6-1128*t^7), t, n+1), t, n), n = 0..20); # G. C. Greubel, Aug 24 2019

Mathematica

CoefficientList[Series[(1+t)*(1-t^6)/(1-48*t+1175*t^6-1128*t^7), {t, 0, 20}], t] (* G. C. Greubel, Sep 15 2017 *)
coxG[{6, 1128, -47}] (* The coxG program is at A169452 *) (* G. C. Greubel, Aug 24 2019 *)

PARI

my(t='t+O('t^20)); Vec((1+t)*(1-t^6)/(1-48*t+1175*t^6-1128*t^7)) \\ G. C. Greubel, Sep 15 2017

Sage

def A164350_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1+t)*(1-t^6)/(1-48*t+1175*t^6-1128*t^7)).list()
A164350_list(20) # G. C. Greubel, Aug 24 2019

Formula

G.f.: (t^6 + 2*t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(1128*t^6 - 47*t^5 - 47*t^4 - 47*t^3 - 47*t^2 - 47*t + 1).

a(n) = -1128*a(n-6) + 47*Sum_{k=1..5} a(n-k). - Wesley Ivan Hurt, May 11 2021