A167943 Number of reduced words of length n in Coxeter group on 28 generators S_i with relations (S_i)^2 = (S_i S_j)^16 = I.

1, 28, 756, 20412, 551124, 14880348, 401769396, 10847773692, 292889889684, 7908027021468, 213516729579636, 5764951698650172, 155653695863554644, 4202649788315975388, 113471544284531335476, 3063731695682346057852

Offset: 0

Author

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

Keywords

• nonn

Comments

The initial terms coincide with those of A170747, 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 (26,26,26,26,26,26,26,26,26,26,26,26,26,26,26,-351).

Crossrefs

Cf. A154638, A169452, A170747.

Programs

Magma

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1+x)*(1-x^16)/(1-27*x+377*x^16-351*x^17) )); // G. C. Greubel, Sep 08 2023

Mathematica

CoefficientList[Series[(1+t)*(1-t^16)/(1-27*t+377*t^16-351*t^17), {t, 0, 50}], t] (* G. C. Greubel, Jul 02 2016; Sep 08 2023 *)
coxG[{16,35,-26}] (* The coxG program is at A169452 *) (* Harvey P. Dale, Mar 20 2021 *)

SageMath

def A167943_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1+x)*(1-x^16)/(1-27*x+377*x^16-351*x^17) ).list()
A167943_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)/( 351*t^16 - 26*t^15 - 26*t^14 - 26*t^13 - 26*t^12 - 26*t^11 - 26*t^10 - 26*t^9 - 26*t^8 - 26*t^7 - 26*t^6 - 26*t^5 - 26*t^4 - 26*t^3 - 26*t^2 - 26*t + 1).

a(n) = -351*a(n-16) + 26*Sum_{k=1..15} a(n-k). - Wesley Ivan Hurt, Sep 03 2022

G.f.: (1+t)*(1-t^16)/(1 - 27*t + 377*t^16 - 351*t^17). - G. C. Greubel, Sep 08 2023