A346802 Number of ways to start with set {1,2,...,n} and then repeat (n+1) times: partition each set into subsets.

1, 1, 4, 35, 561, 14532, 558426, 29947185, 2141867440, 197304236151, 22773405820375, 3221070321954212, 548135428211610344, 110514990079832223628, 26057791266228066121614, 7105134240266115177248187, 2218719629100693497237788887, 786736247267010426995743418575

Offset: 0

Author

Alois P. Heinz, Aug 04 2021

Keywords

• nonn

Comments

Also the number of (n+2)-level labeled rooted trees with n leaves.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..247

Crossrefs

First upper diagonal of A144150.

Cf. A139383, A261280.

Programs

Maple

a:= n-> (g-> coeff(series(1+(g@@(n+2))(x), x, n+1), x, n)*n!)(x-> exp(x)-1):
seq(a(n), n=0..20);
# second Maple program:
A:= proc(n, k) option remember; `if`(n=0 or k=0, 1,
      add(binomial(n-1, j-1)*A(j, k-1)*A(n-j, k), j=1..n))
    end:
a:= n-> A(n, n+1):
seq(a(n), n=0..20);
# third Maple program:
b:= proc(n, t, m) option remember; `if`(n=0, `if`(t=0, 1,
      b(m, t-1, 0)), m*b(n-1, t, m)+b(n-1, t, m+1))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..20);

Mathematica

b[n_, t_, m_] := b[n, t, m] = If[n == 0, If[t == 0, 1, b[m, t - 1, 0]], m*b[n - 1, t, m] + b[n - 1, t, m + 1]];
a[n_] := b[n, n, 0];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Nov 18 2023, after 3rd Maple program *)

Formula

a(n) = n! * [x^n] 1 + g^(n+2)(x), where g(x) = exp(x)-1.

a(n) = A144150(n,n+1).

Conjecture: a(n) ~ c * n^(2*n - 5/6) / (exp(n) * 2^n), where c = 42.345... - Vaclav Kotesovec, Aug 11 2021