A167924 Number of reduced words of length n in Coxeter group on 16 generators S_i with relations (S_i)^2 = (S_i S_j)^16 = I.

1, 16, 240, 3600, 54000, 810000, 12150000, 182250000, 2733750000, 41006250000, 615093750000, 9226406250000, 138396093750000, 2075941406250000, 31139121093750000, 467086816406250000, 7006302246093749880

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 (14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,-105).

Crossrefs

Cf. A154638, A169452, A170735.

Programs

Magma

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1+x)*(1-x^16)/(1-15*x+119*x^16-105*x^17) )); // G. C. Greubel, Sep 10 2023

Mathematica

CoefficientList[Series[(1+t)*(1-t^16)/(1-15*t+119*t^16-105*t^17), {t, 0, 50}], t] (* G. C. Greubel, Jul 01 2016; Sep 10 2023 *)
coxG[{16,105,-14}] (* The coxG program is at A169452 *) (* Harvey P. Dale, Jul 10 2017 *)

SageMath

def A167924_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1+x)*(1-x^16)/(1-15*x+119*x^16-105*x^17) ).list()
A167924_list(40) # G. C. Greubel, Sep 10 2023

Formula

G.f.: (t^16 + 2*t^15 + 2*t^14 + 2*t^13 + 2*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^16 - 14*t^15 - 14*t^14 - 14*t^13 - 14*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, Sep 10 2023: (Start)

G.f.: (1+t)*(1-t^16)/(1 - 15*t + 119*t^16 - 105*t^17).

a(n) = 14*Sum_{j=1..15} a(n-j) - 105*a(n-16). (End)