A165515 Number of reduced words of length n in Coxeter group on 30 generators S_i with relations (S_i)^2 = (S_i S_j)^9 = I.

1, 30, 870, 25230, 731670, 21218430, 615334470, 17844699630, 517496289270, 15007392388395, 435214379250840, 12621216997908960, 366015292928763240, 10614443494626832560, 307818861335266403640, 8926746978464285228160

Offset: 0

Author

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

Keywords

• nonn

Comments

The initial terms coincide with those of A170749, 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..682
• Index entries for linear recurrences with constant coefficients, signature (28,28,28,28,28,28,28,28,-406).

Programs

GAP

a:=[30, 870, 25230, 731670, 21218430, 615334470, 17844699630, 517496289270, 15007392388395];; for n in [10..20] do a[n]:=28*Sum([1..8], j-> a[n-j]) -406*a[n-6]; od; Concatenation([1], a); # G. C. Greubel, Sep 16 2019

Magma

R:=PowerSeriesRing(Integers(), 20); Coefficients(R!( (1+t)*(1-t^9)/(1-29*t+434*t^9-406*t^10) )); // G. C. Greubel, Oct 21 2018

Maple

seq(coeff(series((1+t)*(1-t^9)/(1-29*t+434*t^9-406*t^10), t, n+1), t, n), n = 0 .. 20); # G. C. Greubel, Sep 16 2019

Mathematica

CoefficientList[Series[(1+t)*(1-t^9)/(1-29*t+434*t^9-406*t^10), {t,0,20}], t] (* G. C. Greubel, Oct 21 2018 *)
coxG[{9, 406, -28}] (* 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)/(1-29*t+434*t^9-406*t^10)) \\ G. C. Greubel, Oct 21 2018

Sage

def A165515_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1+t)*(1-t^9)/(1-29*t+434*t^9-406*t^10)).list()
A165515_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)/( 406*t^9 -28*t^8 -28*t^7 -28*t^6 -28*t^5 -28*t^4 -28*t^3 -28*t^2 -28*t + 1).