A164112 Number of reduced words of length n in Coxeter group on 42 generators S_i with relations (S_i)^2 = (S_i S_j)^6 = I.

1, 42, 1722, 70602, 2894682, 118681962, 4865959581, 199504307520, 8179675161840, 335366622329760, 13750029083987280, 563751092750630400, 23113790715369815580, 947665251746544828000, 38854268450681230932000, 1593024724769968897327200, 65314002165544342871757600

Offset

0,2

Comments

The initial terms coincide with those of A170761, 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..615

Index entries for linear recurrences with constant coefficients, signature (40,40,40,40,40,-820).

Formula

G.f.: (t^6 + 2*t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(820*t^6 - 40*t^5 - 40*t^4 - 40*t^3 - 40*t^2 - 40*t + 1).

a(n) = -820*a(n-6) + 40*Sum_{k=1..5} a(n-k). - Wesley Ivan Hurt, May 11 2021

Maple

seq(coeff(series((1+t)*(1-t^6)/(1-41*t+860*t^6-820*t^7), t, n+1), t, n), n = 0 .. 30); # G. C. Greubel, Aug 16 2019

Mathematica

CoefficientList[Series[(1+t)*(1-t^6)/(1-41*t+860*t^6-820*t^7), {t, 0, 30}], t] (* G. C. Greubel, Sep 11 2017 *)
coxG[{6, 820, -40}] (* The coxG program is at A169452 *) (* Harvey P. Dale, Jan 16 2018 *)

Prog

(PARI) my(t='t+O('t^30)); Vec((1+t)*(1-t^6)/(1-41*t+860*t^6-820*t^7)) \\ G. C. Greubel, Sep 11 2017
(Magma) R<t>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+t)*(1-t^6)/(1-41*t+860*t^6-820*t^7) )); // G. C. Greubel, Aug 16 2019
(Sage)
def A164112_list(prec):
    P.<t> = PowerSeriesRing(ZZ, prec)
    return P((1+t)*(1-t^6)/(1-41*t+860*t^6-820*t^7)).list()
A164112_list(30) # G. C. Greubel, Aug 16 2019
(GAP) a:=[42, 1722, 70602, 2894682, 118681962, 4865959581];; for n in [7..30] do a[n]:=40*(a[n-1] +a[n-2]+a[n-3]+a[n-4]+a[n-5]) -820*a[n-6]; od; Concatenation([1], a); # G. C. Greubel, Aug 16 2019

Crossrefs

Sequence in context: A162879 A163225 A163743 * A164686 A165174 A165693

Adjacent sequences: A164109 A164110 A164111 * A164113 A164114 A164115

Keyword

nonn

Author

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

Status

approved