A166603 Number of reduced words of length n in Coxeter group on 21 generators S_i with relations (S_i)^2 = (S_i S_j)^12 = I.

1, 21, 420, 8400, 168000, 3360000, 67200000, 1344000000, 26880000000, 537600000000, 10752000000000, 215040000000000, 4300799999999790, 86015999999991600, 1720319999999748210, 34406399999993288400

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 (19,19,19,19,19,19,19,19,19,19,19,-190).

Crossrefs

Cf. A154638, A169452, A170740.

Programs

Magma

R:=PowerSeriesRing(Integers(), 50); Coefficients(R!( (1+x)*(1-x^12)/(1-20*x+209*x^12-190*x^13) )); // G. C. Greubel, Jan 21 2025

Mathematica

CoefficientList[Series[(1+t)*(1-t^12)/(1-20*t+209*t^12-190*t^13), {t,0,50}], t] (* G. C. Greubel, May 18 2016; Jan 21 2025 *)
coxG[{12,190,-19}] (* The coxG program is at A169452 *) (* G. C. Greubel, Jan 21 2025 *)

SageMath

def A166603_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1+x)*(1-x^12)/(1-20*x+209*x^12-190*x^13) ).list()
A166603_list(50) # G. C. Greubel, Jan 21 2025

Formula

G.f.: (t^12 + 2*t^11 + 2*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^12 - 19*t^11 - 19*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).

From G. C. Greubel, Jan 21 2025: (Start)

a(n) = 19*Sum_{j=1..11} a(n-j) - 190*a(n-12).

G.f.: (1+x)*(1-x^12)/(1 - 20*x + 209*x^12 - 190*x^13). (End)