A163745 Number of reduced words of length n in Coxeter group on 43 generators S_i with relations (S_i)^2 = (S_i S_j)^5 = I.

1, 43, 1806, 75852, 3185784, 133802025, 5619647124, 236023587219, 9912923799660, 416339991317124, 17486161688852682, 734413837213650321, 30845173108213815708, 1295488532304021561975, 54410151353124129064362

Offset: 0

Author

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

Keywords

• nonn

Comments

The initial terms coincide with those of A170762, 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..614
• Index entries for linear recurrences with constant coefficients, signature (41, 41, 41, 41, -861).

Programs

Magma

R:=PowerSeriesRing(Integers(), 20); Coefficients(R!( (1+t)*(1-t^5)/(1-42*t+902*t^5-861*t^6) )); // G. C. Greubel, Aug 09 2019

Maple

seq(coeff(series((1+t)*(1-t^5)/(1-42*t+902*t^5-861*t^6), t, n+1), t, n), n = 0 .. 20); # G. C. Greubel, Aug 09 2019

Mathematica

CoefficientList[Series[(1+t)*(1-t^5)/(1-42*t+902*t^5-861*t^6), {t, 0, 20}], t] (* G. C. Greubel, Aug 02 2017 *)
coxG[{5, 865, -41}] (* 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)/(1-42*t+902*t^5-861*t^6)) \\ G. C. Greubel, Aug 02 2017

Sage

def A163745_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1+t)*(1-t^5)/(1-42*t+902*t^5-861*t^6)).list()
A163745_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)/(861*t^5 - 41*t^4 - 41*t^3 - 41*t^2 - 41*t + 1).

a(n) = 41*a(n-1)+41*a(n-2)+41*a(n-3)+41*a(n-4)-861*a(n-5). - Wesley Ivan Hurt, May 11 2021