A167961 Number of reduced words of length n in Coxeter group on 45 generators S_i with relations (S_i)^2 = (S_i S_j)^16 = I.

1, 45, 1980, 87120, 3833280, 168664320, 7421230080, 326534123520, 14367501434880, 632170063134720, 27815482777927680, 1223881242228817920, 53850774658067988480, 2369434084954991493120, 104255099738019625697280

Offset: 0

Author

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

Keywords

• nonn

Comments

The initial terms coincide with those of A170764, 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 (43,43,43,43,43,43,43,43,43,43, 43,43,43,43,43,-946).

Crossrefs

Cf. A154638, A169452, A170764.

Programs

Magma

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1+x)*(1+x^16)/(1-44*x+946*x^16-903*x^17) )); // G. C. Greubel, Apr 27 2023

Mathematica

CoefficientList[Series[(1+x)*(1+x^16)/(1-44*x+946*x^16-903*x^17), {x, 0, 50}], x] (* G. C. Greubel, Jul 02 2016; Apr 27 2023 *)
coxG[{16, 946, -43, 40}] (* The coxG program is at A169452 *) (* G. C. Greubel, Apr 27 2023 *)

SageMath

def A167961_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1+x)*(1+x^16)/(1-44*x+946*x^16-903*x^17) ).list()
A167961_list(40) # G. C. Greubel, Apr 27 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)/( 946*t^16 - 43*t^15 - 43*t^14 - 43*t^13 - 43*t^12 - 43*t^11 - 43*t^10 - 43*t^9 - 43*t^8 - 43*t^7 - 43*t^6 - 43*t^5 - 43*t^4 - 43*t^3 - 43*t^2 - 43*t + 1).

From G. C. Greubel, Apr 27 2023: (Start)

G.f.: (1 + x)*(1 + x^16)/(1 - 44*x + 946*x^16 - 903*x^17).

a(n) = 43*Sum_{k=1..15} a(n-k) - 946*a(n-16). (End)