A166584 Number of reduced words of length n in Coxeter group on 16 generators S_i with relations (S_i)^2 = (S_i S_j)^12 = I.

1, 16, 240, 3600, 54000, 810000, 12150000, 182250000, 2733750000, 41006250000, 615093750000, 9226406250000, 138396093749880, 2075941406246400, 31139121093669120, 467086816404633600, 7006302246063456000

Offset: 0

Author

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

Keywords

• nonn

Comments

The initial terms coincide with those of A170735, 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 (14,14,14,14,14,14,14,14,14,14,14,-105).

Crossrefs

Cf. A154638, A169452, A170735.

Programs

Magma

R:=PowerSeriesRing(Integers(), 40);
Coefficients(R!( (1+x)*(1-x^12)/(1-15*x+119*x^12-105*x^13) )); // G. C. Greubel, Dec 04 2024

Mathematica

CoefficientList[Series[(1+t)*(1-t^12)/(1-15*t+119*t^12-105*t^13), {t,0,50}], t] (* G. C. Greubel, May 17 2016; Dec 04 2024 *)
coxG[{12,105,-14}] (* The coxG program is at A169452 *) (* G. C. Greubel, Dec 04 2024 *)

SageMath

def A166584_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1+x)*(1-x^12)/(1-15*x+119*x^12-105*x^13) ).list()
A166584_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)/(105*t^12 - 14*t^11 - 14*t^10 - 14*t^9 -14*t^8 - 14*t^7 - 14*t^6 - 14*t^5 - 14*t^4 - 14*t^3 - 14*t^2 -14*t +1).

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

a(n) = 14*Sum_{j=1..11} a(n-j) - 105*a(n-12).

G.f.: (1+x)*(1-x^12)/(1 - 15*x + 119*x^12 - 105*x^13). (End)