A166410 Number of reduced words of length n in Coxeter group on 16 generators S_i with relations (S_i)^2 = (S_i S_j)^11 = I.

1, 16, 240, 3600, 54000, 810000, 12150000, 182250000, 2733750000, 41006250000, 615093750000, 9226406249880, 138396093746400, 2075941406169120, 31139121092133600, 467086816375956000, 7006302245548620000

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,-105).

Crossrefs

Cf. A154638, A169452, A170735.

Programs

Magma

R:=PowerSeriesRing(Integers(), 30);
Coefficients(R!( (1+x)*(1-x^11)/(1-15*x+119*x^11-105*x^12) )); // G. C. Greubel, Jul 23 2024

Mathematica

With[{p=105, q=14}, CoefficientList[Series[(1+t)*(1-t^11)/(1-(q+1)*t + (p+q)*t^11-p*t^12), {t,0,40}], t]] (* G. C. Greubel, May 12 2016; Jul 23 2024 *)
coxG[{11,105,-14}] (* The coxG program is at A169452 *) (* Harvey P. Dale, May 24 2021 *)

SageMath

def A166410_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1+x)*(1-x^11)/(1-15*x+119*x^11-105*x^12) ).list()
A166410_list(30) # G. C. Greubel, Jul 23 2024

Formula

G.f.: (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^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, Jul 23 2024: (Start)

a(n) = 14*Sum_{j=1..10} a(n-j) - 105*a(n-11).

G.f.: (1+x)*(1-x^11)/(1 - 15*x + 119*x^11 - 105*x^12). (End)