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A166518
Number of reduced words of length n in Coxeter group on 7 generators S_i with relations (S_i)^2 = (S_i S_j)^12 = I.
1
1, 7, 42, 252, 1512, 9072, 54432, 326592, 1959552, 11757312, 70543872, 423263232, 2539579371, 15237476100, 91424855865, 548549130780, 3291294758220, 19747768390560, 118486609390800, 710919650629440, 4265517869484480
OFFSET
0,2
COMMENTS
The initial terms coincide with those of A003949, 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)/(15*t^12 - 5*t^11 - 5*t^10 - 5*t^9 - 5*t^8 - 5*t^7 - 5*t^6 - 5*t^5 - 5*t^4 - 5*t^3 - 5*t^2 - 5*t + 1).
From G. C. Greubel, Aug 03 2024: (Start)
a(n) = 5*Sum_{j=1..11} a(n-j) - 15*a(n-12).
G.f.: (1+x)*(1-x^12)/(1 - 6*x + 20*x^12 - 15*x^13). (End)
MATHEMATICA
With[{p=15, q=5}, CoefficientList[Series[(1+t)*(1-t^12)/(1 - (q+1)*t + (p+q)*t^12 - p*t^13), {t, 0, 40}], t]] (* G. C. Greubel, May 15 2016; Aug 03 2024 *)
coxG[{12, 15, -5, 30}] (* The coxG program is at A169452 *) (* G. C. Greubel, Aug 03 2024 *)
PROG
(PARI) Vec((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)/(15*t^12 - 5*t^11 - 5*t^10 - 5*t^9 - 5*t^8 - 5*t^7 - 5*t^6 - 5*t^5 - 5*t^4 - 5*t^3 - 5*t^2 - 5*t + 1)+O(t^99))
(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(15, 5, x) )); // G. C. Greubel, Aug 03 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 A166518_list(prec):
P.<x> = PowerSeriesRing(ZZ, prec)
return P( f(15, 5, x) ).list()
A166518_list(30) # G. C. Greubel, Aug 03 2024
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
John Cannon and N. J. A. Sloane, Dec 03 2009
STATUS
approved