A165777 Number of reduced words of length n in Coxeter group on 6 generators S_i with relations (S_i)^2 = (S_i S_j)^10 = I.

1, 6, 30, 150, 750, 3750, 18750, 93750, 468750, 2343750, 11718735, 58593600, 292967640, 1464836400, 7324173000, 36620820000, 183103875000, 915518250000, 4577585625000, 22887900000000, 114439359375210, 572196093753000, 2860976953154040, 14304867187737000

Offset: 0

Author

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

Keywords

• nonn

Comments

The initial terms coincide with those of A003948, 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..500
• Index entries for linear recurrences with constant coefficients, signature (4,4,4,4,4,4,4,4,4,-10).

Crossrefs

Cf. A003948, A154638.

Programs

GAP

a:=[6, 30, 150, 750, 3750, 18750, 93750, 468750, 2343750, 11718735];; for n in [11..30] do a[n]:=4*Sum([1..9], j-> a[n-j]) -10*a[n-10]; od; Concatenation([1], a); # G. C. Greubel, Sep 17 2019

Magma

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+t)*(1-t^10)/(1-5*t+14*t^10-10*t^11) )); // G. C. Greubel, Sep 17 2019

Maple

A165777 := proc(n)
coeftayl( (t^10 + 2*t^9 + 2*t^8 + 2*t^7 + 2*t^6 + 2*t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(10*t^10 - 4*t^9 - 4*t^8 - 4*t^7 - 4*t^6 - 4*t^5 - 4*t^4 - 4*t^3 - 4*t^2 - 4*t + 1), t=0, n);
end proc:
seq(A165777(n), n=0..25); # Wesley Ivan Hurt, Nov 14 2014

Mathematica

CoefficientList[Series[(t^10 + 2*t^9 + 2*t^8 + 2*t^7 + 2*t^6 + 2*t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(10*t^10 - 4*t^9 - 4*t^8 - 4*t^7 - 4*t^6 - 4*t^5 - 4*t^4 - 4*t^3 - 4*t^2 - 4*t + 1), {t, 0, 25}], t] (* Wesley Ivan Hurt, Nov 14 2014 *)
coxG[{10, 10, -4}] (* The coxG program is at A169452 *) (* G. C. Greubel, Sep 17 2019 *)

PARI

my(t='t+O('t^30)); Vec((1+t)*(1-t^10)/(1-5*t+14*t^10-10*t^11)) \\ G. C. Greubel, Sep 17 2019

Sage

def A165777_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1+t)*(1-t^10)/(1-5*t+14*t^10-10*t^11)).list()
A165777_list(30) # G. C. Greubel, Sep 17 2019

Formula

G.f.: (t^10 + 2*t^9 + 2*t^8 + 2*t^7 + 2*t^6 + 2*t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(10*t^10 - 4*t^9 - 4*t^8 - 4*t^7 - 4*t^6 - 4*t^5 - 4*t^4 - 4*t^3 - 4*t^2 - 4*t + 1).

Extensions

More terms from Wesley Ivan Hurt, Nov 14 2014