A163743 Number of reduced words of length n in Coxeter group on 42 generators S_i with relations (S_i)^2 = (S_i S_j)^5 = I.

1, 42, 1722, 70602, 2894682, 118681101, 4865889840, 199500036960, 8179442209680, 335354699064000, 13749442969516380, 563723074403412000, 23112478470537775200, 947604746561778765600, 38851512911134346287200

Offset: 0

Author

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

Keywords

• nonn

Comments

The initial terms coincide with those of A170761, 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..615
• Index entries for linear recurrences with constant coefficients, signature (40, 40, 40, 40, -820).

Programs

Magma

R:=PowerSeriesRing(Integers(), 20); Coefficients(R!( (1+x)*(1-x^5)/(1-41*x+860*x^5-820*x^6) )); // G. C. Greubel, May 24 2019

Mathematica

CoefficientList[Series[(1+x)*(1-x^5)/(1-41*x+860*x^5-820*x^6), {x,0,20}], x] (* G. C. Greubel, Aug 02 2017 *)
coxG[{5, 820, -40}] (* The coxG program is at A169452 *) (* G. C. Greubel, May 24 2019 *)

PARI

my(x='x+O('x^20)); Vec((1+x)*(1-x^5)/(1-41*x+860*x^5-820*x^6)) \\ G. C. Greubel, Aug 02 2017

Sage

((1+x)*(1-x^5)/(1-41*x+860*x^5-820*x^6)).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, May 24 2019

Formula

G.f.: (t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(820*t^5 - 40*t^4 - 40*t^3 - 40*t^2- 40*t + 1).

a(n) = 40*a(n-1)+40*a(n-2)+40*a(n-3)+40*a(n-4)-820*a(n-5). - Wesley Ivan Hurt, May 11 2021