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A246213
Number of endofunctions on [n] where the largest cycle length equals 3.
2
2, 32, 500, 8600, 165690, 3568768, 85372280, 2251589600, 65007768650, 2041482333440, 69330316507452, 2533173484572640, 99124829660524850, 4137148176815360000, 183498069976613613680, 8620747043700633797888, 427712115490907106172050, 22350263436559575406220800
OFFSET
3,1
LINKS
FORMULA
a(n) ~ (3*exp(11/6)-2*exp(3/2)) * n^(n-1). - Vaclav Kotesovec, Aug 21 2014
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, 3) -A(n, 2):
seq(a(n), n=3..25);
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[n_] := A[n, 3] - A[n, 2];
a /@ Range[3, 25] (* Jean-François Alcover, Dec 28 2020, after Alois P. Heinz *)
CROSSREFS
Column k=3 of A241981.
Sequence in context: A230131 A115418 A163952 * A022028 A013776 A174491
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Aug 19 2014
STATUS
approved