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

0,2

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).

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)

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 *)

Prog

(Magma)
R<x>:=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
(SageMath)
def A166584_list(prec):
    P.<x> = 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

Crossrefs

Cf. A154638, A169452, A170735.

Sequence in context: A165308 A165876 A166410 * A167026 A167117 A167672

Adjacent sequences: A166581 A166582 A166583 * A166585 A166586 A166587

Keyword

nonn

Author

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

Status

approved