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A273996
Number of endofunctions on [n] whose cycle lengths are factorials.
6
1, 1, 4, 25, 218, 2451, 33952, 560407, 10750140, 235118665, 5775676496, 157448312649, 4716609543736, 154007821275595, 5443783515005760, 207093963680817511, 8436365861409555728, 366403740283162634193, 16900793597898691865920, 825115046704241167668025
OFFSET
0,3
LINKS
MAPLE
b:= proc(n) option remember; local r, f, g;
if n=0 then 1 else r, f, g:= $0..2;
while f<=n do r:= r+(f-1)!*b(n-f)*
binomial(n-1, f-1); f, g:= f*g, g+1
od; r fi
end:
a:= n-> add(b(j)*n^(n-j)*binomial(n-1, j-1), j=0..n):
seq(a(n), n=0..20);
MATHEMATICA
b[n_] := b[n] = Module[{r, f, g}, If[n == 0, 1, {r, f, g} = {0, 1, 2}; While[f <= n, r = r + (f - 1)!*b[n - f]*Binomial[n - 1, f - 1]; {f, g} = {f*g, g + 1}]; r]];
a[0] = 1; a[n_] := Sum[b[j]*n^(n - j)*Binomial[n - 1, j - 1], {j, 0, n}];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Jun 06 2018, from Maple *)
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Jun 06 2016
STATUS
approved