A163803 Number of reduced words of length n in Coxeter group on 47 generators S_i with relations (S_i)^2 = (S_i S_j)^5 = I.

1, 47, 2162, 99452, 4574792, 210439351, 9680160420, 445285093005, 20483009107740, 942213581113500, 43341602191631640, 1993703464046530125, 91709888457205975050, 4218633208251709753275, 194056131188825472581550

Offset: 0

Author

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

Keywords

• nonn
• easy

Comments

The initial terms coincide with those of A170766, although the two sequences are eventually different.

Computed with MAGMA using commands similar to those used to compute A154638.

Links

• Colin Barker, Table of n, a(n) for n = 0..600
• Index entries for linear recurrences with constant coefficients, signature (45,45,45,45,-1035).

Programs

GAP

a:=[47,2162,99452,4574792,210439351];; for n in [6..30] do a[n]:=45*(a[n-1]+a[n-2]+a[n-3]+a[n-4]) -1035*a[n-5]; od; Concatenation([1], a); # G. C. Greubel, Aug 09 2019

Magma

R:=PowerSeriesRing(Integers(), 20); Coefficients(R!( (1+t)*(1-t^5)/(1-46*t+1080*t^5-1035*t^6) )); // G. C. Greubel, Aug 09 2019

Maple

seq(coeff(series((1+t)*(1-t^5)/(1-46*t+1080*t^5-1035*t^6), t, n+1), t, n), n = 0 .. 20); # G. C. Greubel, Aug 09 2019

Mathematica

CoefficientList[Series[(1+t)*(1-t^5)/(1-46*t+1080*t^5-1035*t^6), {t, 0, 20}], t] (* G. C. Greubel, Aug 04 2017 *)
coxG[{5, 1035, -45}] (* The coxG program is at A169452 *) (* G. C. Greubel, Aug 09 2019 *)

PARI

my(t='t+O('t^20)); Vec((1+t)*(1-t^5)/(1-46*t+1080*t^5-1035*t^6)) \\ G. C. Greubel, Aug 04 2017

Sage

def A163803_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1+t)*(1-t^5)/(1-46*t+1080*t^5-1035*t^6)).list()
A163803_list(20) # G. C. Greubel, Aug 09 2019

Formula

G.f.: (t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(1035*t^5 - 45*t^4 - 45*t^3 - 45*t^2 - 45*t + 1).

a(n) = 45*a(n-1)+45*a(n-2)+45*a(n-3)+45*a(n-4)-1035*a(n-5). - Wesley Ivan Hurt, May 11 2021