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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
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.
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
KEYWORD
nonn
AUTHOR
John Cannon and N. J. A. Sloane, Dec 03 2009
STATUS
approved