A163837 Number of reduced words of length n in Coxeter group on 50 generators S_i with relations (S_i)^2 = (S_i S_j)^5 = I.

1, 50, 2450, 120050, 5882450, 288238825, 14123642400, 692055537600, 33910577282400, 1661611227897600, 81418604280421800, 3989494661371228800, 195484407940615651200, 9578695296400885468800, 469354075590339325411200

Offset: 0

Author

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

Keywords

• nonn
• easy

Comments

The initial terms coincide with those of A170769, 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..590
• Index entries for linear recurrences with constant coefficients, signature (48,48,48,48,-1176).

Crossrefs

Cf. A154638, A170769.

Programs

GAP

a:=[50,2450,120050,5882450,288238825];; for n in [6..20] do a[n]:=48*(a[n-1]+a[n-2]+a[n-3]+a[n-4]) -1176*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)/(1-49*t+1224*t^5-1176*t^6) )); // G. C. Greubel, Aug 09 2019

Maple

seq(coeff(series((1+t)*(1-t^5)/(1-49*t+1224*t^5-1176*t^6), t, n+1), t, n), n = 0 .. 20); # G. C. Greubel, Aug 09 2019

Mathematica

CoefficientList[Series[(1+t)*(1-t^5)/(1-49*t+1224*t^5-1176*t^6), {t, 0, 20}], t] (* G. C. Greubel, Aug 05 2017 *)
coxG[{5, 1176, -48}] (* The coxG program is at A169452 *) (* G. C. Greubel, Aug 10 2019 *)

PARI

my(t='t+O('t^20)); Vec((1+t)*(1-t^5)/(1-49*t+1224*t^5-1176*t^6)) \\ G. C. Greubel, Aug 05 2017

Sage

def A163748_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1+t)*(1-t^5)/(1-49*t+1224*t^5-1176*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)/(1176*t^5 - 48*t^4 - 48*t^3 - 48*t^2 - 48*t + 1).

a(n) = 48*a(n-1)+48*a(n-2)+48*a(n-3)+48*a(n-4)-1176*a(n-5). - Wesley Ivan Hurt, May 11 2021