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

0,2

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).

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)

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 *)

Prog

(Magma) R<x>:=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
(SageMath)
def A167978_list(prec):
    P.<x> = 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

Crossrefs

Cf. A154638, A169452, A170766.

Sequence in context: A167100 A167644 A167862 * A168724 A168772 A168820

Adjacent sequences: A167975 A167976 A167977 * A167979 A167980 A167981

Keyword

nonn

Author

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

Status

approved