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A222029
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Triangle of number of functions in a size n set for which the sequence of composition powers ends in a length k cycle.
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15
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1, 1, 3, 1, 16, 9, 2, 125, 93, 32, 6, 1296, 1155, 480, 150, 24, 20, 16807, 17025, 7880, 3240, 864, 840, 262144, 292383, 145320, 71610, 24192, 26250, 720, 0, 0, 504, 0, 420, 4782969, 5752131, 3009888, 1692180, 653184, 773920, 46080, 5040, 0, 32256, 0, 26880, 0, 0, 2688
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OFFSET
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0,3
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COMMENTS
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If you take the powers of a finite function you generate a lollipop graph. This table organizes the lollipops by cycle size. The table organized by total lollipop size with the tail included is A225725.
Warning: For T(n,k) after the sixth row there are zero entries and k can be greater than n: T(7,k) = |{1=>262144, 2=>292383, 3=>145320, 4=>71610, 5=>24192, 6=>26250, 7=>720, 8=>0, 9=>0, 10=>504, 11=>0, 12=>420}|.
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LINKS
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FORMULA
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EXAMPLE
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T(1,1) = |{[0]}|, T(2,1) = |{[0,0],[0,1],[1,1]}|, T(2,2) = |{[0,1]}|.
Triangle starts:
1;
1;
3, 1;
16, 9, 2;
125, 93, 32, 6;
1296, 1155, 480, 150, 24, 20;
16807, 17025, 7880, 3240, 864, 840;
262144, 292383, 145320, 71610, 24192, 26250, 720, 0, 0, 504, 0, 420;
...
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MAPLE
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b:= proc(n, m) option remember; `if`(n=0, x^m, add((j-1)!*
b(n-j, ilcm(m, j))*binomial(n-1, j-1), j=1..n))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..degree(p)))(add(
b(j, 1)*n^(n-j)*binomial(n-1, j-1), j=0..n)):
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MATHEMATICA
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b[n_, m_]:=b[n, m]=If[n==0, x^m, Sum[(j - 1)!*b[n - j, LCM[m, j]] Binomial[n - 1, j - 1], {j, n}]]; T[n_]:=If[n==0, {1}, Drop[CoefficientList[Sum[b[j, 1]n^(n - j)*Binomial[n - 1, j - 1], {j, 0, n}], x], 1]]; Table[T[n], {n, 0, 10}]//Flatten (* Indranil Ghosh, Aug 17 2017 *)
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PROG
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#(Ruby 1.9+) see link.
(Python)
from sympy.core.cache import cacheit
from sympy import binomial, Symbol, lcm, factorial as f, Poly, flatten
x=Symbol('x')
@cacheit
def b(n, m): return x**m if n==0 else sum([f(j - 1)*b(n - j, lcm(m, j))*binomial(n - 1, j - 1) for j in range(1, n + 1)])
def T(n): return Poly(sum([b(j, 1)*n**(n - j)*binomial(n - 1, j - 1) for j in range(n + 1)]), x).all_coeffs()[::-1][1:]
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CROSSREFS
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Number of nonzero elements of rows give A009490.
Last elements of rows give A162682.
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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