A165894 Number of reduced words of length n in Coxeter group on 21 generators S_i with relations (S_i)^2 = (S_i S_j)^10 = I.

1, 21, 420, 8400, 168000, 3360000, 67200000, 1344000000, 26880000000, 537600000000, 10751999999790, 215039999991600, 4300799999748210, 86015999993288400, 1720319999832252000, 34406399995974720000

Offset: 0

Author

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

Keywords

• nonn

Comments

The initial terms coincide with those of A170740, 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 (19,19,19,19,19,19,19,19,19,-190).

Crossrefs

Cf. A154638, A170740.

Programs

GAP

a:=[21, 420, 8400, 168000, 3360000, 67200000, 1344000000, 26880000000, 537600000000, 10751999999790];; for n in [7..30] do a[n]:=19*Sum([1..9], j-> a[n-j]) -190*a[n-10]; od; Concatenation([1], a); # G. C. Greubel, Sep 24 2019

Magma

R:=PowerSeriesRing(Integers(), 20); Coefficients(R!( (1+t)*(1-t^10)/(1-20*t+209*t^10-190*t^11) )); // G. C. Greubel, Sep 24 2019

Maple

seq(coeff(series((1+t)*(1-t^10)/(1-20*t+209*t^10-190*t^11), t, n+1), t, n), n = 0..20); # G. C. Greubel, Sep 24 2019

Mathematica

CoefficientList[Series[(1+t)*(1-t^10)/(1-20*t+209*t^10-190*t^11), {t, 0, 20}], t] (* G. C. Greubel, Apr 17 2016 *)
coxG[{10, 190, -19}] (* The coxG program is at A169452 *) (* G. C. Greubel, Sep 24 2019 *)

PARI

my(t='t+O('t^20)); Vec((1+t)*(1-t^10)/(1-20*t+209*t^10-190*t^11)) \\ G. C. Greubel, Sep 24 2019

Sage

def A165894_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1+t)*(1-t^10)/(1-20*t+209*t^10-190*t^11)).list()
A165894_list(30) # G. C. Greubel, Sep 24 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)/(190*t^10 - 19*t^9 - 19*t^8 - 19*t^7 - 19*t^6 - 19*t^5 - 19*t^4 - 19*t^3 - 19*t^2 - 19*t + 1).