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A166495
Number of reduced words of length n in Coxeter group on 5 generators S_i with relations (S_i)^2 = (S_i S_j)^12 = I.
1
1, 5, 20, 80, 320, 1280, 5120, 20480, 81920, 327680, 1310720, 5242880, 20971510, 83886000, 335543850, 1342174800, 5368696800, 21474777600, 85899072000, 343596134400, 1374383923200, 5497533235200, 21990123110400, 87960453120000
OFFSET
0,2
COMMENTS
The initial terms coincide with those of A003947, 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)/(6*t^12 - 3*t^11 - 3*t^10 - 3*t^9 - 3*t^8 - 3*t^7 - 3*t^6 - 3*t^5 - 3*t^4 - 3*t^3 - 3*t^2 - 3*t + 1).
From G. C. Greubel, Aug 02 2024: (Start)
a(n) = 3*Sum_{j=1..11} a(n-j) - 6*a(n-12).
G.f.: (1+x)*(1-x^12)/(1 - 4*x + 9*x^12 - 6*x^13). (End)
MATHEMATICA
CoefficientList[Series[(1+t)*(1-t^12)/(1-4*t+9*t^12-6*t^13), {t, 0, 50}], t] (* G. C. Greubel, May 15 2016; Aug 02 2024 *)
coxG[{12, 6, -3, 30}] (* The coxG program is at A169452 *) (* Harvey P. Dale, Feb 19 2018 *)
PROG
(Magma)
R<x>:=PowerSeriesRing(Integers(), 30);
f:= func< p, q, x | (1+x)*(1-x^12)/(1-(q+1)*x+(p+q)*x^12-p*x^13) >;
Coefficients(R!( f(6, 3, x) )); // G. C. Greubel, Aug 02 2024
(SageMath)
def f(p, q, x): return (1+x)*(1-x^12)/(1-(q+1)*x+(p+q)*x^12-p*x^13)
def A166495_list(prec):
P.<x> = PowerSeriesRing(ZZ, prec)
return P( f(6, 3, x) ).list()
A166495_list(30) # G. C. Greubel, Aug 02 2024
CROSSREFS
KEYWORD
nonn
AUTHOR
John Cannon and N. J. A. Sloane, Dec 03 2009
STATUS
approved