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 A163951 The number of functions in a finite set for which the sequence of composition powers ends in a length 2 cycle. 5
 0, 0, 1, 9, 93, 1155, 17025, 292383, 5752131, 127790505, 3167896005, 86756071545, 2602658092419, 84917405260779, 2994675198208785, 113538315994418175, 4606094297461892895, 199122610252964803857, 9139190793845641425261, 443881600924216704982425 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS The number of functions in a finite set {1,..,n} for which the sequence of composition powers ends in a fixed point gave terms of the sequence A000272(n-1)=(n+1)^(n-1). This is to be seen as a conjecture, and the sequence ending with a length 2 cycle does not seem to have such an easy expression. LINKS Alois P. Heinz, Table of n, a(n) for n = 0..387 FORMULA a(n) ~ (2*exp(3/2)-exp(1)) * n^(n-1). - Vaclav Kotesovec, Aug 20 2014 EXAMPLE Any transposition (or disjoint combination) is one element to be counted. When n=2, there is only one, and a(2)=1. When n=3, there are only 3 transpositions, but there are other 6 elements, for instance f:{1,2,3}->{2,1,1} gives fof:{1,2,3}->{1,2,2} and fofof=f (cycle 2), (the others are similar), thus giving a(3)=9. MAPLE with(combinat): b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,       add((i-1)!^j*multinomial(n, n-i*j, i\$j)/j!*       b(n-i*j, i-1), j=0..n/i)))     end: A:= (n, k)-> add(binomial(n-1, j-1)*n^(n-j)*b(j, min(j, k)), j=0..n): a:= n-> A(n, 2) -A(n, 1): seq(a(n), n=0..25);  # Alois P. Heinz, Aug 19 2014 MATHEMATICA multinomial[n_, k_List] := n!/Times @@ (k!); b[n_, i_] := b[n, i] = If[n == 0, 1, If[i<1, 0, Sum[(i - 1)!^j*multinomial[ n, Join[{n - i*j}, Table[i, j]]]/j!*b[n - i*j, i - 1], {j, 0, n/i}]]]; A[n_, k_] :=  Sum[Binomial[n-1, j-1]*n^(n-j)*b[j, Min[j, k]], {j, 0, n}]; a = 0; a[n_] := A[n, 2] - A[n, 1]; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Jun 05 2018, after Alois P. Heinz *) CROSSREFS Cf. A163947, A163952, A163859. Column k=2 of A222029 and of A241981. Sequence in context: A061635 A261688 A246212 * A231658 A034992 A048359 Adjacent sequences:  A163948 A163949 A163950 * A163952 A163953 A163954 KEYWORD nonn AUTHOR Carlos Alves, Aug 06 2009 EXTENSIONS a(0), a(8)-a(19) added and A246212 merged into this sequence by Alois P. Heinz, Aug 14 2017 STATUS approved

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Last modified September 17 12:57 EDT 2019. Contains 327131 sequences. (Running on oeis4.)