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

0,3

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

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

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 *)

Crossrefs

First upper diagonal of A144150.

Cf. A139383, A261280.

Sequence in context: A334412 A238390 A251591 * A351730 A125798 A129581

Adjacent sequences: A346799 A346800 A346801 * A346803 A346804 A346805

Keyword

nonn

Author

Alois P. Heinz, Aug 04 2021

Status

approved