A350134 Number of endofunctions on [n] with at least one isolated fixed point.

0, 1, 1, 10, 87, 1046, 15395, 269060, 5440463, 124902874, 3208994379, 91208536112, 2841279322871, 96258245162678, 3523457725743059, 138573785311560916, 5827414570508386335, 260928229315498155314, 12393729720071855683739, 622422708333615857463608

Offset: 0

Author

Alois P. Heinz, Dec 15 2021

Keywords

• nonn

Examples

			a(3) = 10: 123, 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 A347999.

Cf. A000312, A001865, A045531, A069856, A204042, A350212.

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..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, n}]];
a[n_] := b[n, 0];
Table[a[n], {n, 0, 23}] (* Jean-François Alcover, Apr 27 2022, after Alois P. Heinz *)

Formula

a(n) = A000312(n) - abs(A069856(n)).

a(n) = Sum_{k=1..n} A350212(n,k).