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

0,4

Links

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

Formula

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

Example

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

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 *)

Crossrefs

Column k=1 of A350212.

Cf. A000035, A000240, A001865, A250105.

Sequence in context: A091708 A327560 A024119 * A120306 A197425 A089379

Adjacent sequences: A348587 A348588 A348589 * A348591 A348592 A348593

Keyword

nonn

Author

Alois P. Heinz, Dec 20 2021

Status

approved