A166600 Number of reduced words of length n in Coxeter group on 19 generators S_i with relations (S_i)^2 = (S_i S_j)^12 = I.

1, 19, 342, 6156, 110808, 1994544, 35901792, 646232256, 11632180608, 209379250944, 3768826516992, 67838877305856, 1221099791505237, 21979796247091188, 395636332447586151, 7121453984055556524

Offset: 0

Author

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

Keywords

• nonn

Comments

The initial terms coincide with those of A170738, 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 (17,17,17,17,17,17,17,17,17,17,17,-153).

Crossrefs

Cf. A154638, A169452, A170738.

Programs

Magma

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1+x)*(1-x^12)/(1 - 18*x+170*x^12-153*x^13) )); // G. C. Greubel, Dec 08 2024

Mathematica

CoefficientList[Series[(1+t)*(1-t^12)/(1-18*t+170*t^12-153*t^13), {t, 0, 50}], t] (* G. C. Greubel, May 18 2016; Dec 08 2024 *)
coxG[{12,153,-17}] (* The coxG program is at A169452 *) (* Harvey P. Dale, Oct 05 2016 *)

SageMath

def A166600_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1+x)*(1-x^12)/(1-18*x+170*x^12-153*x^13) ).list()
print(A166600_list(40)) # G. C. Greubel, Dec 08 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)/(153*t^12 - 17*t^11 - 17*t^10 - 17*t^9 -17*t^8 -17*t^7 - 17*t^6 - 17*t^5 - 17*t^4 - 17*t^3 - 17*t^2 -17*t + 1).

From G. C. Greubel, Dec 08 2024: (Start)

a(n) = 17*Sum_{j=1..11} a(n-j) - 153*a(n-12).

G.f.: (1+x)*(1-x^12)/(1 - 18*x + 170*x^12 - 153*x^13). (End)