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

%I A346802 #17 Nov 18 2023 10:38:30
%S A346802 1,1,4,35,561,14532,558426,29947185,2141867440,197304236151,
%T A346802 22773405820375,3221070321954212,548135428211610344,
%U A346802 110514990079832223628,26057791266228066121614,7105134240266115177248187,2218719629100693497237788887,786736247267010426995743418575
%N A346802 Number of ways to start with set {1,2,...,n} and then repeat (n+1) times: partition each set into subsets.
%C A346802 Also the number of (n+2)-level labeled rooted trees with n leaves.
%H A346802 Alois P. Heinz, <a href="/A346802/b346802.txt">Table of n, a(n) for n = 0..247</a>
%F A346802 a(n) = n! * [x^n] 1 + g^(n+2)(x), where g(x) = exp(x)-1.
%F A346802 a(n) = A144150(n,n+1).
%F A346802 Conjecture: a(n) ~ c * n^(2*n - 5/6) / (exp(n) * 2^n), where c = 42.345... - _Vaclav Kotesovec_, Aug 11 2021
%p A346802 a:= n-> (g-> coeff(series(1+(g@@(n+2))(x), x, n+1), x, n)*n!)(x-> exp(x)-1):
%p A346802 seq(a(n), n=0..20);
%p A346802 # second Maple program:
%p A346802 A:= proc(n, k) option remember; `if`(n=0 or k=0, 1,
%p A346802       add(binomial(n-1, j-1)*A(j, k-1)*A(n-j, k), j=1..n))
%p A346802     end:
%p A346802 a:= n-> A(n, n+1):
%p A346802 seq(a(n), n=0..20);
%p A346802 # third Maple program:
%p A346802 b:= proc(n, t, m) option remember; `if`(n=0, `if`(t=0, 1,
%p A346802       b(m, t-1, 0)), m*b(n-1, t, m)+b(n-1, t, m+1))
%p A346802     end:
%p A346802 a:= n-> b(n$2, 0):
%p A346802 seq(a(n), n=0..20);
%t A346802 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]];
%t A346802 a[n_] := b[n, n, 0];
%t A346802 Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, Nov 18 2023, after 3rd Maple program *)
%Y A346802 First upper diagonal of A144150.
%Y A346802 Cf. A139383, A261280.
%K A346802 nonn
%O A346802 0,3
%A A346802 _Alois P. Heinz_, Aug 04 2021