A166603 Number of reduced words of length n in Coxeter group on 21 generators S_i with relations (S_i)^2 = (S_i S_j)^12 = I.

1, 21, 420, 8400, 168000, 3360000, 67200000, 1344000000, 26880000000, 537600000000, 10752000000000, 215040000000000, 4300799999999790, 86015999999991600, 1720319999999748210, 34406399999993288400

Offset

0,2

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 (19,19,19,19,19,19,19,19,19,19,19,-190).

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)/(190*t^12 - 19*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, Jan 21 2025: (Start)

a(n) = 19*Sum_{j=1..11} a(n-j) - 190*a(n-12).

G.f.: (1+x)*(1-x^12)/(1 - 20*x + 209*x^12 - 190*x^13). (End)

Mathematica

CoefficientList[Series[(1+t)*(1-t^12)/(1-20*t+209*t^12-190*t^13), {t, 0, 50}], t] (* G. C. Greubel, May 18 2016; Jan 21 2025 *)
coxG[{12, 190, -19}] (* The coxG program is at A169452 *) (* G. C. Greubel, Jan 21 2025 *)

Prog

(Magma) R<x>:=PowerSeriesRing(Integers(), 50); Coefficients(R!( (1+x)*(1-x^12)/(1-20*x+209*x^12-190*x^13) )); // G. C. Greubel, Jan 21 2025
(SageMath)
def A166603_list(prec):
    P.<x> = PowerSeriesRing(ZZ, prec)
    return P( (1+x)*(1-x^12)/(1-20*x+209*x^12-190*x^13) ).list()
A166603_list(50) # G. C. Greubel, Jan 21 2025

Crossrefs

Cf. A154638, A169452, A170740.

Sequence in context: A165358 A165894 A166415 * A167074 A167150 A167681

Adjacent sequences: A166600 A166601 A166602 * A166604 A166605 A166606

Keyword

nonn

Author

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

Status

approved