A165876 Number of reduced words of length n in Coxeter group on 16 generators S_i with relations (S_i)^2 = (S_i S_j)^10 = I.

1, 16, 240, 3600, 54000, 810000, 12150000, 182250000, 2733750000, 41006250000, 615093749880, 9226406246400, 138396093669120, 2075941404633600, 31139121063456000, 467086815861120000, 7006302236556000000

Offset: 0

Author

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

Keywords

• nonn

Comments

The initial terms coincide with those of A170735, 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 (14,14,14,14,14,14,14,14,14,-105).

Crossrefs

Cf. A154638, A170735.

Programs

GAP

a:=[16, 240, 3600, 54000, 810000, 12150000, 182250000, 2733750000, 41006250000, 615093749880];; for n in [11..20] do a[n]:=14*Sum([1..9], j-> a[n-j]) -105*a[n-10]; od; Concatenation([1], a); # G. C. Greubel, Aug 10 2019

Magma

R:=PowerSeriesRing(Integers(), 20); Coefficients(R!( (1+t)*(1-t^10)/(1-15*t+129*t^10-105*t^11) )); // G. C. Greubel, Aug 10 2019

Maple

seq(coeff(series((1+t)*(1-t^10)/(1-15*t+129*t^10-105*t^11), t, n+1), t, n), n = 0..20); # G. C. Greubel, Sep 23 2019

Mathematica

CoefficientList[Series[(1+t)*(1-t^10)/(1-15*t+129*t^10-105*t^11), {t, 0, 25}], t] (* G. C. Greubel, Apr 17 2016 *)
coxG[{10, 105, -14}] (* The coxG program is at A169452 *) (* G. C. Greubel, Aug 10 2019 *)

PARI

my(t='t+O('t^20)); Vec((1+t)*(1-t^10)/(1-15*t+129*t^10-105*t^11)) \\ G. C. Greubel, Aug 07 2017

Sage

def A165876_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1+t)*(1-t^10)/(1-15*t+129*t^10-105*t^11)).list()
A165876_list(20) # G. C. Greubel, Aug 10 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)/(105*t^10 - 14*t^9 - 14*t^8 - 14*t^7 - 14*t^6 - 14*t^5 - 14*t^4 - 14*t^3 - 14*t^2 - 14*t + 1).