A163668 Number of reduced words of length n in Coxeter group on 39 generators S_i with relations (S_i)^2 = (S_i S_j)^5 = I.

1, 39, 1482, 56316, 2140008, 81319563, 3090115236, 117423309705, 4462045136796, 169556171182476, 6443075832883092, 244834652131935645, 9303632060115383718, 353534798919570074859, 13434200024194718979990

Offset: 0

Author

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

Keywords

• nonn

Comments

The initial terms coincide with those of A170758, 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..630
• Index entries for linear recurrences with constant coefficients, signature (37, 37, 37, 37, -703).

Programs

GAP

a:=[39, 1482, 56316, 2140008, 81319563];; for n in [6..20] do a[n]:=37*(a[n-1]+a[n-2] +a[n-3]+a[n-4]) -703*a[n-5]; od; Concatenation([1], a); # G. C. Greubel, May 23 2019

Magma

R:=PowerSeriesRing(Integers(), 20); Coefficients(R!( (1+x)*(1-x^5)/(1-38*x+740*x^5-703*x^6) )); // G. C. Greubel, May 23 2019

Mathematica

CoefficientList[Series[(1+x)*(1-x^5)/(1-38*x+740*x^5-703*x^6), {x, 0, 20}], x] (* G. C. Greubel, Aug 01 2017 *)
coxG[{5, 703, -37}] (* The coxG program is at A169452 *) (* G. C. Greubel, May 23 2019 *)

PARI

my(x='x+O('x^20)); Vec((1+x)*(1-x^5)/(1-38*x+740*x^5-703*x^6)) \\ G. C. Greubel, Aug 01 2017

Sage

((1+x)*(1-x^5)/(1-38*x+740*x^5-703*x^6)).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, May 23 2019

Formula

G.f.: (t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(703*t^5 - 37*t^4 - 37*t^3 - 37*t^2 - 37*t + 1).

a(n) = 37*a(n-1)+37*a(n-2)+37*a(n-3)+37*a(n-4)-703*a(n-5). - Wesley Ivan Hurt, May 11 2021