A164030 Number of reduced words of length n in Coxeter group on 31 generators S_i with relations (S_i)^2 = (S_i S_j)^6 = I.

1, 31, 930, 27900, 837000, 25110000, 753299535, 22598972100, 677968744965, 20339049807900, 610171118005500, 18305122253220000, 549153328988465760, 16474589711416216125, 494237386595541683490, 14827112455463543698875

Offset: 0

Author

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

Keywords

• nonn

Comments

The initial terms coincide with those of A170750, 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..675
• Index entries for linear recurrences with constant coefficients, signature (29,29,29,29,29,-435).

Programs

GAP

a:=[31, 930, 27900, 837000, 25110000, 753299535];; for n in [7..30] do a[n]:=29*(a[n-1] +a[n-2]+a[n-3]+a[n-4]+a[n-5]) -435*a[n-6]; od; Concatenation([1], a); # G. C. Greubel, Aug 13 2019

Magma

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+t)*(1-t^6)/(1-30*t+464*t^6-435*t^7) )); // G. C. Greubel, Aug 13 2019

Maple

seq(coeff(series((1+t)*(1-t^6)/(1-30*t+464*t^6-435*t^7), t, n+1), t, n), n = 0 .. 30); # G. C. Greubel, Aug 13 2019

Mathematica

coxG[{6,435,-29}] (* The coxG program is at A169452 *) (* Harvey P. Dale, Sep 29 2016 *)
CoefficientList[Series[(1+t)*(1-t^6)/(1-30*t+464*t^6-435*t^7), {t,0,30}], t] (* G. C. Greubel, Sep 07 2017 *)

PARI

my(t='t+O('t^30)); Vec((1+t)*(1-t^6)/(1-30*t+464*t^6-435*t^7)) \\ G. C. Greubel, Sep 07 2017

Sage

def A164030_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1+t)*(1-t^6)/(1-30*t+464*t^6-435*t^7)).list()
A164030_list(30) # G. C. Greubel, Aug 13 2019

Formula

G.f.: (t^6 + 2*t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(435*t^6 - 29*t^5 - 29*t^4 - 29*t^3 - 29*t^2 - 29*t + 1).

a(n) = -435*a(n-6) + 29*Sum_{k=1..5} a(n-k). - Wesley Ivan Hurt, May 11 2021