A348590 Number of endofunctions on [n] with exactly one isolated fixed point.

0, 1, 0, 9, 68, 845, 12474, 218827, 4435864, 102030777, 2625176150, 74701061831, 2329237613988, 78972674630005, 2892636060014050, 113828236497224355, 4789121681108775344, 214528601554419809777, 10193616586275094959534, 512100888749268955942015

Offset: 0

Author

Alois P. Heinz, Dec 20 2021

Keywords

• nonn

Examples

			a(3) = 9: 122, 133, 132, 121, 323, 321, 113, 223, 213.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..386

Crossrefs

Column k=1 of A350212.

Cf. A000035, A000240, A001865, A250105.

Programs

Maple

g:= proc(n) option remember; add(n^(n-j)*(n-1)!/(n-j)!, j=1..n) end:
b:= proc(n, t) option remember; `if`(n=0, t, add(g(i)*
      b(n-i, `if`(i=1, 1, t))*binomial(n-1, i-1), i=1+t..n))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..23);

Mathematica

g[n_] := g[n] = Sum[n^(n - j)*(n - 1)!/(n - j)!, {j, 1, n}] ;
b[n_, t_] := b[n, t] = If[n == 0, t, Sum[g[i]*
     b[n - i, If[i == 1, 1, t]]*Binomial[n - 1, i - 1], {i, 1 + t, n}]];
a[n_] := b[n, 0];
Table[a[n], {n, 0, 23}] (* Jean-François Alcover, May 16 2022, after Alois P. Heinz *)

Formula

a(n) mod 2 = A000035(n).