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A163952 The number of functions in a finite set for which the sequence of composition powers ends in a length 3 cycle.

%I A163952 #26 Dec 18 2020 20:31:13
%S A163952 0,0,0,2,32,480,7880,145320,3009888,69554240,1779185360,49995179520,
%T A163952 1532580072320,50934256044672,1825145974743000,70172455476381440,
%U A163952 2882264153273207360,125985060813367664640,5840066736661562391968,286204501001426735001600
%N A163952 The number of functions in a finite set for which the sequence of composition powers ends in a length 3 cycle.
%C A163952 See A163951 for the cases ending with length 2 cycles and fixed points.
%H A163952 Alois P. Heinz, <a href="/A163952/b163952.txt">Table of n, a(n) for n = 0..387</a>
%F A163952 a(n) ~ (2*exp(4/3)-exp(1)) * n^(n-1). - _Vaclav Kotesovec_, Aug 18 2017
%e A163952 Any period 3 permutation (or disjoint combinations) is one element to be counted.
%e A163952 For n=3, where there are only 2 cases: f1:{1,2,3}->{2,3,1} and f2:{1,2,3}->{3,1,2} but for n>3 there are other elements (non-permutations) to be counted (for instance, with n=5, we count with f:{1,2,3,4,5}->{2,4,5,3,4}).
%p A163952 b:= proc(n, m) option remember; `if`(m>3, 0, `if`(n=0, x^m, add(
%p A163952       (j-1)!*b(n-j, ilcm(m, j))*binomial(n-1, j-1), j=1..n)))
%p A163952     end:
%p A163952 a:= n-> coeff(add(b(j, 1)*n^(n-j)*binomial(n-1, j-1), j=0..n), x, 3):
%p A163952 seq(a(n), n=0..25);  # _Alois P. Heinz_, Aug 14 2017
%t A163952 b[n_, m_] := b[n, m] = If[m>3, 0, If[n == 0, x^m, Sum[(j - 1)! b[n - j, LCM[m, j]] Binomial[n - 1, j - 1], {j, 1, n}]]];
%t A163952 a[n_] := If[n==0, 0, Coefficient[Sum[b[j, 1] n^(n-j) Binomial[n-1, j-1], {j, 0, n}], x, 3]];
%t A163952 a /@ Range[0, 25] (* _Jean-François Alcover_, Dec 18 2020, after _Alois P. Heinz_ *)
%Y A163952 Cf. A163951, A163947, A163859.
%Y A163952 Column k=3 of A222029.
%K A163952 nonn
%O A163952 0,4
%A A163952 _Carlos Alves_, Aug 07 2009
%E A163952 a(0), a(8)-a(19) from _Alois P. Heinz_, Aug 14 2017