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A166441
Number of reduced words of length n in Coxeter group on 47 generators S_i with relations (S_i)^2 = (S_i S_j)^11 = I.
1
1, 47, 2162, 99452, 4574792, 210440432, 9680259872, 445291954112, 20483429889152, 942237774900992, 43342937645445632, 1993775131690497991, 91713656057762857860, 4218828178657089175245, 194066096218225996890780
OFFSET
0,2
COMMENTS
The initial terms coincide with those of A170766, although the two sequences are eventually different.
Computed with MAGMA using commands similar to those used to compute A154638.
LINKS
Index entries for linear recurrences with constant coefficients, signature (45,45,45,45,45,45,45,45,45,45,-1035).
FORMULA
G.f.: (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)/(1035*t^11 - 45*t^10 - 45*t^9 - 45*t^8 - 45*t^7 - 45*t^6 - 45*t^5 - 45*t^4 - 45*t^3 - 45*t^2 - 45*t + 1).
From G. C. Greubel, Jul 26 2024: (Start)
a(n) = 45*Sum_{j=1..10} a(n-j) - 1035*a(n-11).
G.f.: (1+x)*(1-x^11)/(1 - 46*x + 1080*x^11 - 1035*x^12). (End)
MATHEMATICA
coxG[{11, 1035, -45}] (* The coxG program is at A169452 *) (* Harvey P. Dale, Jan 16 2015 *)
With[{p=1035, q=45}, CoefficientList[Series[(1+t)*(1-t^11)/(1 - (q+1)*t + (p+q)*t^11 - p*t^12), {t, 0, 40}], t]] (* G. C. Greubel, May 14 2016; Jul 26 2024 *)
PROG
(Magma)
R<x>:=PowerSeriesRing(Integers(), 30);
f:= func< p, q, x | (1+x)*(1-x^11)/(1-(q+1)*x+(p+q)*x^11-p*x^12) >;
Coefficients(R!( f(1035, 45, x) )); // G. C. Greubel, Jul 26 2024
(SageMath)
def f(p, q, x): return (1+x)*(1-x^11)/(1-(q+1)*x+(p+q)*x^11-p*x^12)
def A166441_list(prec):
P.<x> = PowerSeriesRing(ZZ, prec)
return P( f(1035, 45, x) ).list()
A166441_list(30) # G. C. Greubel, Jul 26 2024
CROSSREFS
KEYWORD
nonn
AUTHOR
John Cannon and N. J. A. Sloane, Dec 03 2009
STATUS
approved