A167980 Number of reduced words of length n in Coxeter group on 48 generators S_i with relations (S_i)^2 = (S_i S_j)^16 = I.

1, 48, 2256, 106032, 4983504, 234224688, 11008560336, 517402335792, 24317909782224, 1142941759764528, 53718262708932816, 2524758347319842352, 118663642324032590544, 5577191189229531755568, 262127985893787992511696

Offset: 0

Author

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

Keywords

• nonn

Comments

The initial terms coincide with those of A170767, 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 (46,46,46,46,46,46,46,46,46,46,46,46,46,46,46,-1081).

Crossrefs

Cf. A154638, A169452, A170767.

Programs

Magma

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+x)*(1-x^17)/(1-47*x+1127*x^16-1081*x^17) )); // G. C. Greubel, Jan 17 2023

Mathematica

coxG[{16,1081,-46}] (* The coxG program is at A169452 *) (* Harvey P. Dale, Jun 12 2016 *)
CoefficientList[Series[(1+t)*(1-t^17)/(1-47*t+1127*t^16-1081*t^17), {t, 0,50}], t] (* G. C. Greubel, Jul 03 2016; Jan 17 2023 *)

SageMath

def A167980_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1+x)*(1-x^17)/(1-47*x+1127*x^16-1081*x^17) ).list()
A167980_list(30) # G. C. Greubel, Jan 17 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)/( 1081*t^16 - 46*t^15 - 46*t^14 - 46*t^13 - 46*t^12 - 46*t^11 - 46*t^10 - 46*t^9 - 46*t^8 - 46*t^7 - 46*t^6 - 46*t^5 - 46*t^4 - 46*t^3 - 46*t^2 - 46*t + 1).

From G. C. Greubel, Jan 17 2023: (Start)

a(n) = Sum_{j=1..15} a(n-j) - 1081*a(n-16).

G.f.: (1+x)*(1-x^17)/(1 - 47*x + 1127*x^16 - 1081*x^17). (End)