A163391 Number of reduced words of length n in Coxeter group on 9 generators S_i with relations (S_i)^2 = (S_i S_j)^5 = I.

1, 9, 72, 576, 4608, 36828, 294336, 2352420, 18801216, 150264576, 1200956652, 9598382640, 76712967828, 613111567824, 4900159716480, 39163451657148, 313005296651040, 2501626174048260, 19993698450611424, 159795249138713664

Offset: 0

Author

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

Keywords

• nonn

Comments

The initial terms coincide with those of A003951, 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..1000
• Index entries for linear recurrences with constant coefficients, signature (7, 7, 7, 7, -28).

Programs

Magma

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+x)*(1-x^5)/(1-8*x+35*x^5-28*x^6) )); // G. C. Greubel, May 12 2019

Mathematica

CoefficientList[Series[(1+x)*(1-x^5)/(1-8*x+35*x^5-28*x^6), {x, 0, 30}], x] (* or *) LinearRecurrence[{7,7,7,7,-28}, {1,9,72,576,4608,36828}, 30] (* G. C. Greubel, Dec 21 2016 *)
coxG[{5, 28, -7}] (* The coxG program is at A169452 *) (* G. C. Greubel, May 12 2019 *)

PARI

my(x='x+O('x^30)); Vec((1+x)*(1-x^5)/(1-8*x+35*x^5-28*x^6)) \\ G. C. Greubel, Dec 21 2016

Sage

((1+x)*(1-x^5)/(1-8*x+35*x^5-28*x^6)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, May 12 2019

Formula

G.f.: (t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(28*t^5 - 7*t^4 - 7*t^3 - 7*t^2 - 7*t + 1).

a(n) = 7*a(n-1)+7*a(n-2)+7*a(n-3)+7*a(n-4)-28*a(n-5). - Wesley Ivan Hurt, May 10 2021