A166415 Number of reduced words of length n in Coxeter group on 21 generators S_i with relations (S_i)^2 = (S_i S_j)^11 = I.

1, 21, 420, 8400, 168000, 3360000, 67200000, 1344000000, 26880000000, 537600000000, 10752000000000, 215039999999790, 4300799999991600, 86015999999748210, 1720319999993288400, 34406399999832252000

Offset: 0

Author

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

Keywords

• nonn

Comments

The initial terms coincide with those of A170740, 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 (19,19,19,19,19,19,19,19,19,19,-190).

Crossrefs

Cf. A154638, A169452, A170740.

Programs

Magma

R:=PowerSeriesRing(Integers(), 30);
Coefficients(R!( (1+x)*(1-x^11)/(1-20*x+209*x^11-190*x^12) )); // G. C. Greubel, Jul 23 2024

Mathematica

CoefficientList[Series[(1+t)*(1-t^11)/(1-20*t+209*t^11-190*t^12), {t, 0, 50}], t] (* G. C. Greubel, May 13 2016; Jul 23 2024 *)
coxG[{11, 190, -19, 30}] (* The coxG program is at A169452 *) (* G. C. Greubel, Jul 23 2024 *)

SageMath

def A166415_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1+x)*(1-x^11)/(1-20*x+209*x^11-190*x^12) ).list()
A166415_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)/(190*t^11 - 19*t^10 - 19*t^9 - 19*t^8 - 19*t^7 - 19*t^6 - 19*t^5 - 19*t^4 - 19*t^3 - 19*t^2 - 19*t + 1).

From G. C. Greubel, Jul 23 2024: (Start)

a(n) = 19*Sum_{j=1..10} a(n-j) - 190*a(n-11).

G.f.: (1+x)*(1-x^11)/(1 - 20*x + 209*x^11 - 190*x^12). (End)