A168684 Number of reduced words of length n in Coxeter group on 7 generators S_i with relations (S_i)^2 = (S_i S_j)^17 = I.

1, 7, 42, 252, 1512, 9072, 54432, 326592, 1959552, 11757312, 70543872, 423263232, 2539579392, 15237476352, 91424858112, 548549148672, 3291294892032, 19747769352171, 118486616112900, 710919696676665, 4265518180055580

Offset: 0

Author

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

Keywords

• nonn

Comments

The initial terms coincide with those of A003949, although the two sequences are eventually different.

First disagreement at index 17: a(17) = 19747769352171, A003949(17) = 19747769352192. - 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 (5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,-15).

Crossrefs

Cf. A003949 (g.f.: (1+x)/(1-6*x)).

Programs

Magma

R:=PowerSeriesRing(Integers(), 40);
Coefficients(R!( (1+t)*(1-t^17)/(1 -6*t +20*t^17 -15*t^18) )); // G. C. Greubel, Mar 24 2021

Mathematica

CoefficientList[Series[(1+t)*(1-t^17)/(1 -6*t +20*t^17 -15*t^18), {t, 0, 50}], t] (* G. C. Greubel, Aug 03 2016; Mar 24 2021 *)
coxG[{17, 15, -5, 40}] (* The coxG program is at A169452 *) (* G. C. Greubel, Mar 24 2021 *)

Sage

def A168684_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1+t)*(1-t^17)/(1 -6*t +20*t^17 -15*t^18) ).list()
A168684_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)/ (15*t^17 - 5*t^16 - 5*t^15 - 5*t^14 - 5*t^13 - 5*t^12 - 5*t^11 - 5*t^10 - 5*t^9 - 5*t^8 - 5*t^7 - 5*t^6 - 5*t^5 - 5*t^4 - 5*t^3 - 5*t^2 - 5*t + 1).

G.f.: (1+t)*(1-t^17)/(1 - 6*t + 20*t^17 - 15*t^18). - G. C. Greubel, Mar 24 2021