A166414 Number of reduced words of length n in Coxeter group on 20 generators S_i with relations (S_i)^2 = (S_i S_j)^11 = I.

1, 20, 380, 7220, 137180, 2606420, 49521980, 940917620, 17877434780, 339671260820, 6453753955580, 122621325155830, 2329805177957160, 44266298381117640, 841059669239935560, 15980133715534083240

Offset

0,2

Comments

The initial terms coincide with those of A170739, 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 (18,18,18,18,18,18,18,18,18,18,-171).

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)/(171*t^11 - 18*t^10 - 18*t^9 - 18*t^8 - 18*t^7 - 18*t^6 - 18*t^5 - 18*t^4 - 18*t^3 - 18*t^2 - 18*t + 1).

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

a(n) = 18*Sum_{j=1..10} a(n-j) - 171*a(n-11).

G.f.: (1+x)*(1 - x^11)/(1 - 19*x + 189*x^11 - 171*x^12). (End)

Mathematica

coxG[{11, 171, -18}] (* The coxG program is at A169452 *) (* Harvey P. Dale, Sep 07 2015 *)
CoefficientList[Series[(1+t)*(1-t^11)/(1-19*t+189*t^11-171*t^12), {t, 0, 50}], t] (* G. C. Greubel, May 12 2016; Jul 23 2024 *)

Prog

(Magma)
R<x>:=PowerSeriesRing(Integers(), 30);
Coefficients(R!( (1+x)*(1-x^11)/(1-19*x+189*x^11-171*x^12) )); // G. C. Greubel, Jul 23 2024
(SageMath)
def A166414_list(prec):
    P.<x> = PowerSeriesRing(ZZ, prec)
    return P( (1+x)*(1-x^11)/(1-19*x+189*x^11-171*x^12) ).list()
A166414_list(30) # G. C. Greubel, Jul 23 2024
(PARI) Vec((1+x+x^2+x^3+x^4+x^5+x^6+x^7+x^8+x^9+x^10)*(1+x)/(1-18*x-18*x^2-18*x^3-18*x^4-18*x^5-18*x^6-18*x^7-18*x^8-18*x^9-18*x^10+171*x^11)+O(x^99)) \\ Charles R Greathouse IV, Jun 08 2026
(PARI) a(n)=if(n, ([0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0; 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0; 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0; 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0; 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0; 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0; 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0; 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0; 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0; 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1; -171, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18]^(n-1)*[20; 380; 7220; 137180; 2606420; 49521980; 940917620; 17877434780; 339671260820; 6453753955580; 122621325155830])[1, 1], 1) \\ Charles R Greathouse IV, Jun 08 2026

Crossrefs

Cf. A154638, A169452, A170739.

Sequence in context: A165348 A165882 A063815 * A166601 A167073 A167148

Adjacent sequences: A166411 A166412 A166413 * A166415 A166416 A166417

Keyword

nonn,easy

Author

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

Status

approved