A168687 Number of reduced words of length n in Coxeter group on 10 generators S_i with relations (S_i)^2 = (S_i S_j)^17 = I.

1, 10, 90, 810, 7290, 65610, 590490, 5314410, 47829690, 430467210, 3874204890, 34867844010, 313810596090, 2824295364810, 25418658283290, 228767924549610, 2058911320946490, 18530201888518365, 166771816996664880

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.

First disagreement at index 17: a(17) = 18530201888518365, A003952(17) = 18530201888518410. - Klaus Brockhaus, Mar 30 2011

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 (8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,-36).

Crossrefs

Cf. A003952 (g.f.: (1+x)/(1-9*x)).

Programs

Magma

R:=PowerSeriesRing(Integers(), 40);
Coefficients(R!( (1+t)*(1-t^17)/(1 -9*t +44*t^17 -36*t^18) )); // G. C. Greubel, Mar 24 2021

Mathematica

coxG[{17,36,-8}] (* The coxG program is at A169452 *) (* Harvey P. Dale, Dec 10 2015 *)
CoefficientList[Series[(1+t)*(1-t^17)/(1 -9*t +44*t^17 -36*t^18), {t, 0, 50}], t] (* G. C. Greubel, Aug 03 2016; Mar 24 2021 *)

Sage

def A168687_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1+t)*(1-t^17)/(1 -9*t +44*t^17 -36*t^18) ).list()
A168687_list(40) # G. C. Greubel, Mar 24 2021

Formula

G.f.: (t^17 + 2*t^16 + 2*t^15 + 2*t^14 + 2*t^13 + 2*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^17 - 8*t^16 - 8*t^15 - 8*t^14 - 8*t^13 - 8*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).

G.f.: (1+t)*(1-t^17)/(1 -9 *t + 44*t^17 - 36*t^18). - G. C. Greubel, Mar 24 2021