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A246215
Number of endofunctions on [n] where the largest cycle length equals 5.
2
24, 864, 24696, 688128, 19840464, 604727424, 19632956112, 680195957760, 25130679950376, 988325574652416, 41277744231187464, 1826323584590389248, 85391029667937905184, 4209030460729215184896, 218223423136426488339744, 11875233973816788160610304
OFFSET
5,1
LINKS
FORMULA
a(n) ~ (5*exp(137/60) - 4*exp(25/12)) * 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, 5) -A(n, 4):
seq(a(n), n=5..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, 5] - A[n, 4];
a /@ Range[5, 25] (* Jean-François Alcover, Dec 28 2020, after Alois P. Heinz *)
CROSSREFS
Column k=5 of A241981.
Sequence in context: A275563 A275088 A291111 * A109575 A160111 A227666
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Aug 19 2014
STATUS
approved