A164070 Number of reduced words of length n in Coxeter group on 37 generators S_i with relations (S_i)^2 = (S_i S_j)^6 = I.

1, 37, 1332, 47952, 1726272, 62145792, 2237247846, 80540898480, 2899471482810, 104380942332240, 3757712806199520, 135277620783782400, 4869992899598197770, 175319692235303773500, 6311507043063165819750

Offset

0,2

Comments

The initial terms coincide with those of A170756, 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..640

Index entries for linear recurrences with constant coefficients, signature (35,35,35,35,35,-630).

Formula

G.f.: (t^6 + 2*t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(630*t^6 - 35*t^5 - 35*t^4 - 35*t^3 - 35*t^2 - 35*t + 1).

a(n) = -630*a(n-6) + 35*Sum_{k=1..5} a(n-k). - Wesley Ivan Hurt, May 05 2021

Maple

seq(coeff(series((1+t)*(1-t^6)/(1-36*t+665*t^6-630*t^7), t, n+1), t, n), n = 0 .. 30); # G. C. Greubel, Aug 16 2019

Mathematica

CoefficientList[Series[(1+t)*(1-t^6)/(1-36*t+665*t^6-630*t^7), {t, 0, 30}], t] (* G. C. Greubel, Sep 09 2017 *)
coxG[{6, 630, -35}] (* The coxG program is at A169452 *) (* G. C. Greubel, Aug 16 2019 *)

Prog

(PARI) my(t='t+O('t^30)); Vec((1+t)*(1-t^6)/(1-36*t+665*t^6-630*t^7)) \\ G. C. Greubel, Sep 09 2017
(Magma) R<t>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+t)*(1-t^6)/(1-36*t+665*t^6-630*t^7) )); // G. C. Greubel, Aug 16 2019
(Sage)
def A164070_list(prec):
    P.<t> = PowerSeriesRing(ZZ, prec)
    return P((1+t)*(1-t^6)/(1-36*t+665*t^6-630*t^7)).list()
A164070_list(30) # G. C. Greubel, Aug 16 2019
(GAP) a:=[37, 1332, 47952, 1726272, 62145792, 2237247846];; for n in [7..30] do a[n]:=35*(a[n-1] +a[n-2]+a[n-3]+a[n-4]+a[n-5]) -630*a[n-6]; od; Concatenation([1], a); # G. C. Greubel, Aug 16 2019

Crossrefs

Sequence in context: A162851 A163220 A163645 * A164673 A165169 A165654

Adjacent sequences: A164067 A164068 A164069 * A164071 A164072 A164073

Keyword

nonn

Author

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

Status

approved