A167942 Number of reduced words of length n in Coxeter group on 27 generators S_i with relations (S_i)^2 = (S_i S_j)^16 = I.

1, 27, 702, 18252, 474552, 12338352, 320797152, 8340725952, 216858874752, 5638330743552, 146596599332352, 3811511582641152, 99099301148669952, 2576581829865418752, 66991127576500887552, 1741769316989023076352

Offset: 0

Author

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

Keywords

• nonn

Comments

The initial terms coincide with those of A170746, 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 (25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,-325).

Crossrefs

Cf. A154638, A169452, A170758.

Programs

Magma

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1+x)*(1-x^16)/(1-26*x+350*x^16-325*x^17) )); // G. C. Greubel, Sep 08 2023

Mathematica

CoefficientList[Series[(1+t)*(1-t^16)/(1-26*t+350*t^16-325*t^17), {t, 0, 50}], t] (* G. C. Greubel, Jul 01 2016; Sep 08 2023 *)
coxG[{16,325,-25}] (* The coxG program is at A169452 *) (* Harvey P. Dale, Oct 28 2018 *)

SageMath

def A167942_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1+x)*(1-x^16)/(1-26*x+350*x^16-325*x^17) ).list()
A167942_list(40) # G. C. Greubel, Sep 08 2023

Formula

G.f.: (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)/( 325*t^16 - 25*t^15 - 25*t^14 - 25*t^13 - 25*t^12 - 25*t^11 - 25*t^10 - 25*t^9 - 25*t^8 - 25*t^7 - 25*t^6 - 25*t^5 - 25*t^4 - 25*t^3 - 25*t^2 - 25*t + 1).

From G. C. Greubel, Sep 08 2023: (Start)

G.f.: (1+t)*(1-t^16)/(1 - 26*t + 350*t^16 - 325*t^17).

a(n) = 25*Sum_{j=1..15} a(n-j) - 325*a(n-16). (End)