A165757 Number of reduced words of length n in Coxeter group on 5 generators S_i with relations (S_i)^2 = (S_i S_j)^10 = I.

1, 5, 20, 80, 320, 1280, 5120, 20480, 81920, 327680, 1310710, 5242800, 20971050, 83883600, 335532000, 1342118400, 5368435200, 21473587200, 85893734400, 343572480000, 1374280089690, 5497081037820, 21988166868630, 87952038348420

Offset

0,2

Comments

The initial terms coincide with those of A003947, 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 (3,3,3,3,3,3,3,3,3,-6).

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

Maple

seq(coeff(series((1+t)*(1-t^10)/(1-4*t+9*t^10-6*t^11), t, n+1), t, n), n = 0..30); # G. C. Greubel, Sep 17 2019

Mathematica

CoefficientList[Series[(1+t)*(1-t^10)/(1-4*t+9*t^10-6*t^11), {t, 0, 30}], t] (* G. C. Greubel, Apr 07 2016 *)
coxG[{10, 6, -3}] (* 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)/(1-4*t+9*t^10-6*t^11)) \\ G. C. Greubel, Sep 17 2019
(Magma) R<t>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+t)*(1-t^10)/(1-4*t+9*t^10-6*t^11) )); // G. C. Greubel, Sep 17 2019
(Sage)
def A165757_list(prec):
    P.<t> = PowerSeriesRing(ZZ, prec)
    return P((1+t)*(1-t^10)/(1-4*t+9*t^10-6*t^11)).list()
A165757_list(30) # G. C. Greubel, Sep 17 2019
(GAP) a:=[5, 20, 80, 320, 1280, 5120, 20480, 81920, 327680, 1310710];; for n in [11..30] do a[n]:=3*Sum([1..9], j-> a[n-j]) -6*a[n-10]; od; Concatenation([1], a); # G. C. Greubel, Sep 17 2019

Crossrefs

Sequence in context: A164354 A164706 A165185 * A166331 A166495 A166859

Adjacent sequences: A165754 A165755 A165756 * A165758 A165759 A165760

Keyword

nonn

Author

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

Status

approved