A166585 Number of reduced words of length n in Coxeter group on 17 generators S_i with relations (S_i)^2 = (S_i S_j)^12 = I.

1, 17, 272, 4352, 69632, 1114112, 17825792, 285212672, 4563402752, 73014444032, 1168231104512, 18691697672192, 299067162754936, 4785074604076800, 76561193665194120, 1224979098642551040, 19599665578271938560

Offset: 0

Author

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

Keywords

• nonn

Comments

The initial terms coincide with those of A170736, 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 (15,15,15,15,15,15,15,15,15,15,15,-120).

Crossrefs

Cf. A154638, A169452, A170736.

Programs

Magma

R:=PowerSeriesRing(Integers(), 40);
Coefficients(R!( (1+x)*(1-x^12)/(1-16*x+135*x^12-120*x^13) )); // G. C. Greubel, Dec 04 2024

Mathematica

CoefficientList[Series[(1+t)*(1-t^12)/(1-16*t+135*t^12-120*t^13), {t, 0, 50}], t] (* G. C. Greubel, May 17 2016; Dec 04 2024 *)
coxG[{16,1081,-46}] (* The coxG program is at A169452 *) (* G. C. Greubel, Dec 04 2024 *)

SageMath

def A166585_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1+x)*(1-x^12)/(1-16*x+135*x^12-120*x^13) ).list()
A166585_list(40) # G. C. Greubel, Dec 04 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)/(120*t^12 - 15*t^11 - 15*t^10 - 15*t^9 -15*t^8 - 15*t^7 - 15*t^6 - 15*t^5 - 15*t^4 - 15*t^3 - 15*t^2 -15*t +1).

From G. C. Greubel, Dec 04 2024: (Start)

a(n) = 15*Sum_{j=1..11} a(n-j) - 120*a(n-12).

G.f.: (1+x)*(1-x^12)/(1 - 16*x + 135*x^12 - 120*x^13). (End)