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

0,2

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).

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).

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 *)

Prog

(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
(Magma) R<t>:=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
(Sage)
def A165777_list(prec):
    P.<t> = 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
(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

Crossrefs

Cf. A003948, A154638.

Sequence in context: A164365 A164741 A165213 * A166364 A166500 A166877

Adjacent sequences: A165774 A165775 A165776 * A165778 A165779 A165780

Keyword

nonn

Author

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

Extensions

More terms from Wesley Ivan Hurt, Nov 14 2014

Status

approved