A165782 Number of reduced words of length n in Coxeter group on 7 generators S_i with relations (S_i)^2 = (S_i S_j)^10 = I.

1, 7, 42, 252, 1512, 9072, 54432, 326592, 1959552, 11757312, 70543851, 423262980, 2539577145, 15237458460, 91424724300, 548548187040, 3291288169680, 19747723302720, 118486305524160, 710917627392000, 4265504529834660

Offset: 0

Author

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

Keywords

• nonn

Comments

The initial terms coincide with those of A003949, 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..500
• Index entries for linear recurrences with constant coefficients, signature (5,5,5,5,5,5,5,5,5,-15).

Programs

GAP

a:=[7, 42, 252, 1512, 9072, 54432, 326592, 1959552, 11757312, 70543851];; for n in [11..30] do a[n]:=5*Sum([1..9], j-> a[n-j]) -15*a[n-10]; od; Concatenation([1], a); # G. C. Greubel, Sep 22 2019

Magma

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+t)*(1-t^10)/(1-6*t+15*t^10-6*t^11) )); // G. C. Greubel, Sep 22 2019

Maple

seq(coeff(series((1+t)*(1-t^10)/(1-6*t+15*t^10-6*t^11), t, n+1), t, n), n = 0 .. 30); # G. C. Greubel, Sep 22 2019

Mathematica

CoefficientList[Series[(1+t)*(1-t^10)/(1-6*t+15*t^10-6*t^11), {t, 0, 30}], t] (* G. C. Greubel, Apr 08 2016 *)
coxG[{10, 15, -5}] (* The coxG program is at A169452 *) (* G. C. Greubel, Sep 22 2019 *)

PARI

my(t='t+O('t^30)); Vec((1+t)*(1-t^10)/(1-6*t+15*t^10-6*t^11)) \\ G. C. Greubel, Aug 07 2017

Sage

def A165782_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1+t)*(1-t^10)/(1-6*t+15*t^10-6*t^11) ).list()
A165782_list(30) # G. C. Greubel, Sep 22 2019

Formula

G.f.: (t^10 + 2*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)/(15*t^10 - 5*t^9 - 5*t^8 - 5*t^7 - 5*t^6 - 5*t^5 - 5*t^4 - 5*t^3 - 5*t^2 - 5*t + 1).