A163748 Number of reduced words of length n in Coxeter group on 44 generators S_i with relations (S_i)^2 = (S_i S_j)^5 = I.

1, 44, 1892, 81356, 3498308, 150426298, 6468290136, 278134727640, 11959718115576, 514264646533176, 22113240807047082, 950863378003793100, 40886868257711476308, 1758124284303633320844, 75598869044590717310100

Offset: 0

Author

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

Keywords

• nonn

Comments

The initial terms coincide with those of A170763, 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..610
• Index entries for linear recurrences with constant coefficients, signature (42,42,42,42,-903).

Programs

GAP

a:=[44, 1892, 81356, 3498308, 150426298];; for n in [6..30] do a[n]:=42*(a[n-1]+a[n-2]+a[n-3] +a[n-4]) -903*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)/(1-43*t+945*t^5-903*t^6) )); // G. C. Greubel, Aug 09 2019

Maple

seq(coeff(series((1+t)*(1-t^5)/(1-43*t+945*t^5-903*t^6), t, n+1), t, n), n = 0 .. 20); # G. C. Greubel, Aug 09 2019

Mathematica

CoefficientList[Series[(1+t)*(1-t^5)/(1-43*t+945*t^5-903*t^6), {t, 0, 20}], t] (* G. C. Greubel, Aug 02 2017 *)
coxG[{5, 903, -42}] (* 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)/(1-43*t+945*t^5-903*t^6)) \\ G. C. Greubel, Aug 02 2017

Sage

def A163748_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1+t)*(1-t^5)/(1-43*t+945*t^5-903*t^6)).list()
A163748_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)/(903*t^5 - 42*t^4 - 42*t^3 - 42*t^2 - 42*t + 1).

a(n) = 42*a(n-1)+42*a(n-2)+42*a(n-3)+42*a(n-4)-903*a(n-5). - Wesley Ivan Hurt, May 11 2021