A165788 Number of reduced words of length n in Coxeter group on 10 generators S_i with relations (S_i)^2 = (S_i S_j)^10 = I.

1, 10, 90, 810, 7290, 65610, 590490, 5314410, 47829690, 430467210, 3874204845, 34867843200, 313810585200, 2824295234400, 25418656818000, 228767908737600, 2058911155018800, 18530200182592800, 166771799730147600

Offset: 0

Author

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

Keywords

• nonn

Comments

The initial terms coincide with those of A003952, 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 (8,8,8,8,8,8,8,8,8,-36).

Programs

GAP

a:=[10, 90, 810, 7290, 65610, 590490, 5314410, 47829690, 430467210, 3874204845];; for n in [11..20] do a[n]:=8*Sum([1..9], j-> a[n-j]) -36*a[n-10]; od; Concatenation([1], a); # G. C. Greubel, Sep 22 2019

Magma

R:=PowerSeriesRing(Integers(), 20); Coefficients(R!( (1+t)*(1-t^10)/(1-9*t+44*t^10-36*t^11) )); // G. C. Greubel, Sep 22 2019

Maple

seq(coeff(series((1+t)*(1-t^10)/(1-9*t+44*t^10-36*t^11), t, n+1), t, n), n = 0..20); # G. C. Greubel, Aug 10 2019

Mathematica

CoefficientList[Series[(1+t)*(1-t^10)/(1-9*t+44*t^10-36*t^11), {t, 0, 25}], t] (* G. C. Greubel, Apr 08 2016 *)
coxG[{10, 36, -8}] (* The coxG program is at A169452 *) (* G. C. Greubel, Sep 22 2019 *)

PARI

my(t='t+O('t^20)); Vec((1+t)*(1-t^10)/(1-9*t+44*t^10-36*t^11)) \\ G. C. Greubel, Sep 22 2019

Sage

def A165788_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1+t)*(1-t^10)/(1-9*t+44*t^10-36*t^11)).list()
A165788_list(20) # 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)/(36*t^10 - 8*t^9 - 8*t^8 - 8*t^7 - 8*t^6 - 8*t^5 - 8*t^4 - 8*t^3 - 8*t^2 - 8*t + 1).