A163923 Number of reduced words of length n in Coxeter group on 7 generators S_i with relations (S_i)^2 = (S_i S_j)^6 = I.

1, 7, 42, 252, 1512, 9072, 54411, 326340, 1957305, 11739420, 70410060, 422301600, 2532857460, 15191434125, 91114353750, 546480693675, 3277652052150, 19658522431800, 117906811965600, 707175035973000, 4241455800274875

Offset

0,2

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 (5,5,5,5,5,-15).

Formula

G.f.: (t^6 + 2*t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(15*t^6 - 5*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)+5*a(n-5)-15*a(n-6). - Wesley Ivan Hurt, Apr 23 2021

Maple

seq(coeff(series((1+t)*(1-t^6)/(1-6*t+20*t^6-15*t^7), t, n+1), t, n), n = 0 .. 30); # G. C. Greubel, Aug 10 2019

Mathematica

coxG[{6, 15, -5}] (* The coxG program is at A169452 *) (* Harvey P. Dale, Aug 18 2015 *)
CoefficientList[Series[(1+t)*(1-t^6)/(1-6*t+20*t^6-15*t^7), {t, 0, 30}], t] (* G. C. Greubel, Aug 08 2017 *)

Prog

(PARI) my(t='t+O('t^30)); Vec((1+t)*(1-t^6)/(1-6*t+20*t^6-15*t^7)) \\ G. C. Greubel, Aug 08 2017
(Magma) R<t>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+t)*(1-t^6)/(1-6*t+20*t^6-15*t^7) )); // G. C. Greubel, Aug 10 2019
(Sage)
def A163923_list(prec):
    P.<t> = PowerSeriesRing(ZZ, prec)
    return P((1+t)*(1-t^6)/(1-6*t+20*t^6-15*t^7)).list()
A163923_list(30) # G. C. Greubel, Aug 10 2019
(GAP) a:=[7, 42, 252, 1512, 9072, 54411];; for n in [7..30] do a[n]:=5*(a[n-1] +a[n-2]+a[n-3]+a[n-4]+a[n-5]) -15*a[n-6]; od; Concatenation([1], a); # G. C. Greubel, Aug 10 2019

Crossrefs

Sequence in context: A162941 A094168 A163345 * A164369 A164742 A165214

Adjacent sequences: A163920 A163921 A163922 * A163924 A163925 A163926

Keyword

nonn

Author

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

Status

approved