A165456 Number of reduced words of length n in Coxeter group on 28 generators S_i with relations (S_i)^2 = (S_i S_j)^9 = I.

1, 28, 756, 20412, 551124, 14880348, 401769396, 10847773692, 292889889684, 7908027021090, 213516729559224, 5764951697823864, 155653695833814360, 4202649787312378584, 113471544252017775096, 3063731694658235867448

Offset: 0

Author

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

Keywords

• nonn
• easy

Comments

The initial terms coincide with those of A170747, 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..695
• Index entries for linear recurrences with constant coefficients, signature (26,26,26,26,26,26,26,26,-351).

Programs

GAP

a:=[28, 756, 20412, 551124, 14880348, 401769396, 10847773692, 292889889684, 7908027021090];; for n in [10..20] do a[n]:=326*Sum([1..8], j-> a[n-j]) -351*a[n-9]; od; Concatenation([1], a); # G. C. Greubel, Sep 16 2019

Magma

R:=PowerSeriesRing(Integers(), 20); Coefficients(R!( (1+t)*(1-t^9)/(1-27*t+377*t^9-351*t^10) )); // G. C. Greubel, Oct 20 2018

Maple

seq(coeff(series((x^9+2*x^8+2*x^7+2*x^6+2*x^5+2*x^4+2*x^3+2*x^2+2*x+1)/( 351*x^9-26*x^8-26*x^7-26*x^6-26*x^5-26*x^4-26*x^3-26*x^2-26*x+1),x, n+1), x, n), n = 0 .. 15); # Muniru A Asiru, Oct 21 2018

Mathematica

CoefficientList[Series[(1+t)*(1-t^9)/(1-27*t+377*t^9-351*t^10), {t, 0, 30}], t] (* G. C. Greubel, Oct 20 2018 *)
coxG[{9, 351, -26}] (* The coxG program is at A169452 *) (* G. C. Greubel, Sep 16 2019 *)

PARI

my(t='t+O('t^20)); Vec((1+t)*(1-t^9)/(1-27*t+377*t^9-351*t^10)) \\ G. C. Greubel, Oct 20 2018

Sage

def A165456_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1+t)*(1-t^9)/(1-27*t+377*t^9-351*t^10)).list()
A165456_list(20) # G. C. Greubel, Sep 16 2019

Formula

G.f.: (t^9 + 2*t^8 + 2*t^7 + 2*t^6 + 2*t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t +1)/(351*t^9 - 26*t^8 - 26*t^7 - 26*t^6 - 26*t^5 - 26*t^4 - 26*t^3 -26*t^2 - 26*t + 1).