A163345 Number of reduced words of length n in Coxeter group on 7 generators S_i with relations (S_i)^2 = (S_i S_j)^5 = I.

1, 7, 42, 252, 1512, 9051, 54180, 324345, 1941660, 11623500, 69582660, 416548125, 2493614550, 14927719275, 89362970550, 534960522600, 3202475913000, 19171231408875, 114766238286000, 687034086094125, 4112845750671000

Offset: 0

Author

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

Keywords

• nonn
• easy

Comments

The initial terms coincide with those of A003949, 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 (5, 5, 5, 5, -15).

Programs

Magma

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+x)*(1-x^5)/(1-6*x+20*x^5-15*x^6) )); // G. C. Greubel, May 12 2019

Mathematica

CoefficientList[Series[(1+x)*(1-x^5)/(1-6*x+20*x^5-15*x^6), {x, 0, 30}], x] (* or *) LinearRecurrence[{5,5,5,5,-15}, {1,7,42,252,1512,9051}, 30] (* G. C. Greubel, Dec 19 2016 *)
coxG[{5,15,-5}] (* The coxG program is at A169452 *) (* Harvey P. Dale, May 09 2018 *)

PARI

my(x='x+O('x^30)); Vec((1+x)*(1-x^5)/(1-6*x+20*x^5-15*x^6)) \\ G. C. Greubel, Dec 19 2016

Sage

((1+x)*(1-x^5)/(1-6*x+20*x^5-15*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)/(15*t^5 - 5*t^4 - 5*t^3 - 5*t^2 - 5*t + 1).

a(n) = 5*a(n-1)+5*a(n-2)+5*a(n-3)+5*a(n-4)-15*a(n-5). - Wesley Ivan Hurt, May 10 2021