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A164365
Number of reduced words of length n in Coxeter group on 6 generators S_i with relations (S_i)^2 = (S_i S_j)^7 = I.
1
1, 6, 30, 150, 750, 3750, 18750, 93735, 468600, 2342640, 11711400, 58548000, 292695000, 1463250000, 7315125210, 36570003000, 182821904040, 913968987000, 4569142377000, 22842199635000, 114193439625000, 570879418872060
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.
FORMULA
G.f.: (t^7 + 2*t^6 + 2*t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(10*t^7 - 4*t^6 - 4*t^5 - 4*t^4 - 4*t^3 - 4*t^2 - 4*t + 1).
a(n) = -10*a(n-7) + 4*Sum_{k=1..6} a(n-k). - Wesley Ivan Hurt, May 11 2021
MAPLE
seq(coeff(series((1+t)*(1-t^7)/(1-5*t+14*t^7-10*t^8), t, n+1), t, n), n = 0 .. 30); # G. C. Greubel, Aug 28 2019
MATHEMATICA
CoefficientList[Series[(1+t)*(1-t^7)/(1-5*t+14*t^7-10*t^8), {t, 0, 30}], t] (* G. C. Greubel, Sep 15 2017 *)
coxG[{7, 10, -4}] (* The coxG program is at A169452 *) (* G. C. Greubel, Aug 28 2019 *)
PROG
(PARI) my(t='t+O('t^30)); Vec((1+t)*(1-t^7)/(1-5*t+14*t^7-10*t^8)) \\ G. C. Greubel, Sep 15 2017
(Magma) R<t>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+t)*(1-t^7)/(1-5*t+14*t^7-10*t^8) )); // G. C. Greubel, Aug 28 2019
(Sage)
def A164365_list(prec):
P.<t> = PowerSeriesRing(ZZ, prec)
return P((1+t)*(1-t^7)/(1-5*t+14*t^7-10*t^8)).list()
A164365_list(30) # G. C. Greubel, Aug 28 2019
(GAP) a:=[6, 30, 150, 750, 3750, 18750, 93735];; for n in [8..30] do a[n]:=4*(a[n-1] +a[n-2]+a[n-3]+a[n-4]+a[n-5]+a[n-6]) -10*a[n-7]; od; Concatenation([1], a); # G. C. Greubel, Aug 28 2019
CROSSREFS
Sequence in context: A163317 A342807 A163922 * A164741 A165213 A165777
KEYWORD
nonn
AUTHOR
John Cannon and N. J. A. Sloane, Dec 03 2009
STATUS
approved