A166543 Number of reduced words of length n in Coxeter group on 10 generators S_i with relations (S_i)^2 = (S_i S_j)^12 = I.

1, 10, 90, 810, 7290, 65610, 590490, 5314410, 47829690, 430467210, 3874204890, 34867844010, 313810596045, 2824295364000, 25418658272400, 228767924419200, 2058911319481200, 18530201872706400, 166771816830738000

Offset: 0

Author

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

Keywords

• nonn

Comments

The initial terms coincide with those of A003952, 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 (8,8,8,8,8,8,8,8,8,8,8,-36).

Crossrefs

Cf. A003952, A154638, A169452.

Programs

Magma

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+x)*(1-x^12)/(1-9*x+44*x^12-36*x^13) )); // G. C. Greubel, Aug 23 2024

Mathematica

CoefficientList[Series[(1+t)*(1-t^12)/(1-9*t+44*t^12-36*t^13), {t, 0, 50}], t] (* G. C. Greubel, May 16 2016; Aug 23 2024 *)
coxG[{12,36,-8,30}] (* The coxG program is at A169452 *) (* G. C. Greubel, Aug 23 2024 *)

SageMath

def A166543_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1+x)*(1-x^12)/(1-9*x+44*x^12-36*x^13) ).list()
A166543_list(30) # G. C. Greubel, Aug 23 2024

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)/(36*t^12 - 8*t^11 - 8*t^10 - 8*t^9 - 8*t^8 - 8*t^7 - 8*t^6 - 8*t^5 - 8*t^4 - 8*t^3 - 8*t^2 - 8*t + 1).

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

a(n) = 8*Sum_{j=1..11} a(n-j) - 36*a(n-12).

G.f.: (1+x)*(1-x^12)/(1 - 9*x + 44*x^12 - 36*x^13). (End)