A165880 Number of reduced words of length n in Coxeter group on 18 generators S_i with relations (S_i)^2 = (S_i S_j)^10 = I.

1, 18, 306, 5202, 88434, 1503378, 25557426, 434476242, 7386096114, 125563633938, 2134581776793, 36287890202880, 616894133404896, 10487200267134144, 178282404528545952, 3030800876768794752, 51523614901389241440

Offset: 0

Author

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

Keywords

• nonn

Comments

The initial terms coincide with those of A170737, 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 (16,16,16,16,16,16,16,16,16,-136).

Crossrefs

Cf. A154638, A170737.

Programs

GAP

a:=[18, 306, 5202, 88434, 1503378, 25557426, 434476242, 7386096114, 125563633938, 2134581776793];; for n in [11..20] do a[n]:=16*Sum([1..9], j-> a[n-j]) -136*a[n-10]; od; Concatenation([1], a); # G. C. Greubel, Sep 24 2019

Magma

R:=PowerSeriesRing(Integers(), 20); Coefficients(R!( (1+t)*(1-t^10)/(1-17*t+152*t^10-136*t^11) )); // G. C. Greubel, Sep 24 2019

Maple

seq(coeff(series((1+t)*(1-t^10)/(1-17*t+152*t^10-136*t^11), t, n+1), t, n), n = 0..20); # G. C. Greubel, Sep 24 2019

Mathematica

CoefficientList[Series[(1+t)*(1-t^10)/(1-17*t+152*t^10-136*t^11), {t, 0, 20}], t] (* G. C. Greubel, Apr 17 2016 *)
coxG[{10,136,-16}] (* The coxG program is at A169452 *) (* Harvey P. Dale, Nov 04 2017 *)

PARI

my(t='t+O('t^20)); Vec((1+t)*(1-t^10)/(1-17*t+152*t^10-136*t^11)) \\ G. C. Greubel, Sep 24 2019

Sage

def A165880_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1+t)*(1-t^10)/(1-17*t+152*t^10-136*t^11)).list()
A165880_list(20) # G. C. Greubel, Sep 24 2019

Formula

G.f.: (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)/(136*t^10 - 16*t^9 - 16*t^8 - 16*t^7 - 16*t^6 - 16*t^5 - 16*t^4 - 16*t^3 - 16*t^2 - 16*t + 1).