A164685 Number of reduced words of length n in Coxeter group on 41 generators S_i with relations (S_i)^2 = (S_i S_j)^7 = I.

1, 41, 1640, 65600, 2624000, 104960000, 4198400000, 167935999180, 6717439934400, 268697596064820, 10747903790145600, 429916149507936000, 17196645896401920000, 687865832499456000000, 27514633165713408671580

Offset

0,2

Comments

The initial terms coincide with those of A170760, 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..620

Index entries for linear recurrences with constant coefficients, signature (39,39,39,39,39,39,-780).

Formula

G.f.: (t^7 + 2*t^6 + 2*t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(780*t^7 - 39*t^6 - 39*t^5 - 39*t^4 - 39*t^3 - 39*t^2 - 39*t + 1).

Maple

seq(coeff(series((1+t)*(1-t^7)/(1-40*t+819*t^7-780*t^8), t, n+1), t, n), n = 0..20); # G. C. Greubel, Sep 15 2019

Mathematica

CoefficientList[Series[(1+t)*(1-t^7)/(1-40*t+819*t^7-780*t^8), {t, 0, 20}], t] (* G. C. Greubel, Sep 15 2019 *)
coxG[{7, 780, -39}] (* The coxG program is at A169452 *) (* G. C. Greubel, Sep 15 2019 *)

Prog

(PARI) my(t='t+O('t^20)); Vec((1+t)*(1-t^7)/(1-40*t+819*t^7-780*t^8)) \\ G. C. Greubel, Sep 15 2019
(Magma) R<t>:=PowerSeriesRing(Integers(), 20); Coefficients(R!( (1+t)*(1-t^7)/(1-40*t+819*t^7-780*t^8) )); // G. C. Greubel, Sep 15 2019
(Sage)
def A164685_list(prec):
    P.<t> = PowerSeriesRing(ZZ, prec)
    return P((1+t)*(1-t^7)/(1-40*t+819*t^7-780*t^8)).list()
A164685_list(20) # G. C. Greubel, Sep 15 2019
(GAP) a:=[41, 1640, 65600, 2624000, 104960000, 4198400000, 167935999180];; for n in [8..20] do a[n]:=39*(a[n-1] +a[n-2]+a[n-3]+a[n-4]+a[n-5]+a[n-6]) -780*a[n-7]; od; Concatenation([1], a); # G. C. Greubel, Sep 15 2019

Crossrefs

Sequence in context: A163224 A163677 A164091 * A165173 A165692 A166225

Adjacent sequences: A164682 A164683 A164684 * A164686 A164687 A164688

Keyword

nonn

Author

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

Status

approved