A167978 Number of reduced words of length n in Coxeter group on 47 generators S_i with relations (S_i)^2 = (S_i S_j)^16 = I.

1, 47, 2162, 99452, 4574792, 210440432, 9680259872, 445291954112, 20483429889152, 942237774900992, 43342937645445632, 1993775131690499072, 91713656057762957312, 4218828178657096036352, 194066096218226417672192

Offset: 0

Author

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

Keywords

• nonn

Comments

The initial terms coincide with those of A170766, 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 (45,45,45,45,45,45,45,45,45,45,45,45,45,45,45,-1035).

Crossrefs

Cf. A154638, A169452, A170766.

Programs

Magma

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+x)*(1-x^17)/(1-46*x+1081*x^16-1035*x^17) )); // G. C. Greubel, Jan 17 2023

Mathematica

CoefficientList[Series[(1+t)*(1-t^16)/(1-46*t+1081*t^16-1035*t^17), {t, 0,50}], t] (* G. C. Greubel, Jul 03 2016; Jan 17 2023 *)
coxG[{16,1035,-45}] (* The coxG program is at A169452 *) (* G. C. Greubel, Jan 17 2023 *)

SageMath

def A167978_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1+x)*(1-x^17)/(1-46*x+1081*x^16-1035*x^17) ).list()
A167978_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)/( 1035*t^16 - 45*t^15 - 45*t^14 - 45*t^13 - 45*t^12 - 45*t^11 - 45*t^10 - 45*t^9 - 45*t^8 - 45*t^7 - 45*t^6 - 45*t^5 - 45*t^4 - 45*t^3 - 45*t^2 - 45*t + 1).

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

a(n) = Sum_{j=1..15} a(n-j) - 1035*a(n-16).

G.f.: (1+x)*(1-x^16)/(1 - 46*x + 1081*x^16 - 1035*x^17). (End)