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A166463
Number of reduced words of length n in Coxeter group on 50 generators S_i with relations (S_i)^2 = (S_i S_j)^11 = I.
1
1, 50, 2450, 120050, 5882450, 288240050, 14123762450, 692064360050, 33911153642450, 1661646528480050, 81420679895522450, 3989613314880598825, 195491052429149282400, 9579061569028311897600, 469374016882387138922400
OFFSET
0,2
COMMENTS
The initial terms coincide with those of A170769, 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 (48,48,48,48,48,48,48,48,48,48,-1176).
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)/(1176*t^11 - 48*t^10 - 48*t^9 - 48*t^8 - 48*t^7 - 48*t^6 - 48*t^5 - 48*t^4 - 48*t^3 - 48*t^2 - 48*t + 1).
From G. C. Greubel, Jul 27 2024: (Start)
a(n) = 48*Sum_{j=1..10} a(n-j) - 1176*a(n-11).
G.f.: (1+x)*(1-x^11)/(1 - 49*x + 1224*x^11 - 1176*x^12). (End)
MATHEMATICA
With[{p=1176, q=48}, 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 15 2016; Jul 27 2024 *)
coxG[{11, 1176, -48}] (* The coxG program is at A169452 *) (* Harvey P. Dale, Apr 29 2018 *)
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(1176, 48, x) )); // G. C. Greubel, Jul 27 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 A166463_list(prec):
P.<x> = PowerSeriesRing(ZZ, prec)
return P( f(1176, 48, x) ).list()
A166463_list(30) # G. C. Greubel, Jul 27 2024
CROSSREFS
KEYWORD
nonn
AUTHOR
John Cannon and N. J. A. Sloane, Dec 03 2009
STATUS
approved