A167989 Number of reduced words of length n in Coxeter group on 50 generators S_i with relations (S_i)^2 = (S_i S_j)^16 = I.

1, 50, 2450, 120050, 5882450, 288240050, 14123762450, 692064360050, 33911153642450, 1661646528480050, 81420679895522450, 3989613314880600050, 195491052429149402450, 9579061569028320720050, 469374016882387715282450

Offset: 0

Author

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

Keywords

• nonn

Comments

The initial terms coincide with those of A170769, 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 (48,48,48,48,48,48,48,48,48,48,48,48,48,48,48,-1176).

Crossrefs

Cf. A154638, A169452, A170769.

Programs

Magma

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1+x)*(1-x^16)/(1-49*x+1224*x^16-1176*x^17) )); // G. C. Greubel, Jan 14 2023

Mathematica

CoefficientList[Series[(1+x)*(1-x^16)/(1-49*x+1224*x^16-1176*x^17), {x, 0, 50}], x] (* G. C. Greubel, Jul 03 2016; Jan 14 2023 *)
coxG[{16, 1176, -48, 10}] (* The coxG program is at A169452 *) (* G. C. Greubel, Jan 14 2023 *)

SageMath

def A167989_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1+x)*(1-x^16)/(1-49*x+1224*x^16-1176*x^17) ).list()
A167989_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)/( 1176*t^16 - 48*t^15 - 48*t^14 - 48*t^13 - 48*t^12 - 48*t^11 - 48*t^10 - 48*t^9 - 48*t^8 - 48*t^7 - 48*t^6 - 48*t^5 - 48*t^4 - 48*t^3 - 48*t^2 - 48*t + 1).

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

a(n) = -1176*a(n-16) + 48*Sum_{j=1..15} a(n-j).

G.f.: (1+x)*(1-x^16)/(1-49*x+1224*x^16-1176*x^17). (End)