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A166538
Number of reduced words of length n in Coxeter group on 8 generators S_i with relations (S_i)^2 = (S_i S_j)^12 = I.
1
1, 8, 56, 392, 2744, 19208, 134456, 941192, 6588344, 46118408, 322828856, 2259801992, 15818613916, 110730297216, 775112079168, 5425784544768, 37980491747520, 265863441771648, 1861044089174592, 13027308601633536
OFFSET
0,2
COMMENTS
The initial terms coincide with those of A003950, although the two sequences are eventually different.
Computed with MAGMA using commands similar to those used to compute A154638.
LINKS
FORMULA
G.f.: (t^12 + 2*t^11 + 2*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)/(21*t^12 - 6*t^11 - 6*t^10 - 6*t^9 - 6*t^8 - 6*t^7 - 6*t^6 - 6*t^5 - 6*t^4 - 6*t^3 - 6*t^2 - 6*t + 1).
From G. C. Greubel, Aug 23 2024: (Start)
a(n) = 6*Sum_{j=1..11} a(n-j) - 21*a(n-12).
G.f.: (1 + x)*(1 - x^12)/(1 - 7*x + 27*x^12 - 21*x^13). (End)
MATHEMATICA
CoefficientList[Series[(1+t)*(1-t^12)/(1-7*t+27*t^12-21*t^13), {t, 0, 50}], t] (* G. C. Greubel, May 16 2016; Aug 23 2024 *)
coxG[{12, 21, -6}] (* The coxG program is at A169452 *) (* Harvey P. Dale, Oct 24 2016 *)
PROG
(Magma)
R<x>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+x)*(1-x^12)/(1-7*x+27*x^12-21*x^13) )); // G. C. Greubel, Aug 23 2024
(SageMath)
def A166538_list(prec):
P.<x> = PowerSeriesRing(ZZ, prec)
return P( (1+x)*(1-x^12)/(1-7*x+27*x^12-21*x^13) ).list()
A166538_list(30) # G. C. Greubel, Aug 23 2024
CROSSREFS
KEYWORD
nonn
AUTHOR
John Cannon and N. J. A. Sloane, Dec 03 2009
STATUS
approved