A166601 Number of reduced words of length n in Coxeter group on 20 generators S_i with relations (S_i)^2 = (S_i S_j)^12 = I.

1, 20, 380, 7220, 137180, 2606420, 49521980, 940917620, 17877434780, 339671260820, 6453753955580, 122621325156020, 2329805177964190, 44266298381316000, 841059669244935600, 15980133715652476800

Offset: 0

Author

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

Keywords

• nonn

Comments

The initial terms coincide with those of A170739, 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 (18,18,18,18,18,18,18,18,18,18,18,-171).

Crossrefs

Cf. A154638, A169452, A170739.

Programs

Magma

R:=PowerSeriesRing(Integers(), 50); Coefficients(R!( (1+x)*(1-x^12)/(1-19*x+189*x^12-171*x^13) )); // G. C. Greubel, Dec 30 2024

Mathematica

CoefficientList[Series[(1+t)*(1-t^12)/(1-19*t+189*t^12-171*t^13), {t,0,50}], t] (* G. C. Greubel, May 18 2016; Dec 30 2024 *)
coxG[{12,171,-18}] (* The coxG program is at A169452 *) (* G. C. Greubel, Dec 30 2024 *)

SageMath

def A166601_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1+x)*(1-x^12)/(1-19*x+189*x^12-171*x^13) ).list()
A166601_list(50) # G. C. Greubel, Dec 30 2024

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)/(171*t^12 - 18*t^11 - 18*t^10 - 18*t^9 -18*t^8 -18*t^7 - 18*t^6 - 18*t^5 - 18*t^4 - 18*t^3 - 18*t^2 -18*t + 1).

From G. C. Greubel, Dec 30 2024: (Start)

a(n) = 18*Sum_{j=1..11} a(n-j) - 171*a(n-12).

G.f.: (1+x)*(1-x^12)/(1 - 19*x + 189*x^12 - 171*x^13). (End)