A167988 Number of reduced words of length n in Coxeter group on 49 generators S_i with relations (S_i)^2 = (S_i S_j)^16 = I.

1, 49, 2352, 112896, 5419008, 260112384, 12485394432, 599298932736, 28766348771328, 1380784741023744, 66277667569139712, 3181328043318706176, 152703746079297896448, 7329779811806299029504, 351829430966702353416192

Offset: 0

Author

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

Keywords

• nonn

Comments

The initial terms coincide with those of A170768, 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 (47,47,47,47,47,47,47,47,47,47,47,47,47,47,47,-1128).

Crossrefs

Cf. A154638, A169452, A170768.

Programs

Magma

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1+x)*(1-x^16)/(1-48*x+1175*x^16-1128*x^17) )); // G. C. Greubel, Jan 14 2023

Mathematica

coxG[{16,1128,-47}] (* The coxG program is at A169452 *) (* Harvey P. Dale, May 05 2015 *)
CoefficientList[Series[(1+x)*(1-x^16)/(1-48*x+1175*x^16-1128*x^17), {x, 0, 50}], x] (* G. C. Greubel, Jul 03 2016; Jan 14 2023 *)

Sage

def A167988_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1+x)*(1-x^16)/(1-48*x+1175*x^16-1128*x^17) ).list()
A167988_list(40) # G. C. Greubel, Jan 14 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)/( 1128*t^16 - 47*t^15 - 47*t^14 - 47*t^13 - 47*t^12 - 47*t^11 - 47*t^10 - 47*t^9 - 47*t^8 - 47*t^7 - 47*t^6 - 47*t^5 - 47*t^4 - 47*t^3 - 47*t^2 - 47*t + 1).

From G. C. Greubel, Jan 14 2023: (Start)

a(n) = -1128*a(n-16) + 47*Sum_{j=1..15} a(n-j).

G.f.: (1 + x)*(1 - x^16)/(1 - 48*x + 1175*x^16 - 1128*x^17). (End)