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

0,4

Links

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

Formula

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

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

From Mélika Tebni, Jun 10 2026: (Start)

E.g.f.: (1-exp(-x)) / (1+LambertW(-x)).

Limit_{n->oo} a(n) / n^n = 1 - exp(-1/e) = A397014. (End)

Example

a(3) = 10: 123, 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..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 *)

Prog

(Python)
from math import comb
def a_list(N: int) -> list[int]:
    K = [0] * (N + 1)
    F = [1] * (N + 1)
    for i in range(1, N + 1):
        F[i] = F[i-1] * i
    for i in range(1, N + 1):
        val = 0
        p = 1
        for k in range(i):
            val += p * (F[i-1] // F[k])
            p *= i
        K[i] = val
    C = [0] * (N + 1); C[0] = 1
    A = [0] * (N + 1); A[0] = 0
    for n in range(1, N + 1):
        c_n = 0
        for i in range(1, n + 1):
            c_n += K[i] * comb(n - 1, i - 1) * C[n - i]
        C[n] = c_n
        a_n = C[n - 1]
        for i in range(2, n + 1):
            a_n += K[i] * comb(n - 1, i - 1) * A[n - i]
        A[n] = a_n
    return A[:N]
print(a_list(20))  # Peter Luschny, Jun 13 2026

Crossrefs

Column k=1 of A347999.

Cf. A000312, A001865, A045531, A069856, A204042, A350212, A397014.

Sequence in context: A121115 A292998 A114648 * A217417 A136864 A099789

Adjacent sequences: A350131 A350132 A350133 * A350135 A350136 A350137

Keyword

nonn,changed

Author

Alois P. Heinz, Dec 15 2021

Status

approved