A346802 - OEIS

%I #17 Nov 18 2023 10:38:30

%S 1,1,4,35,561,14532,558426,29947185,2141867440,197304236151,

%T 22773405820375,3221070321954212,548135428211610344,

%U 110514990079832223628,26057791266228066121614,7105134240266115177248187,2218719629100693497237788887,786736247267010426995743418575

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

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

%H Alois P. Heinz, <a href="/A346802/b346802.txt">Table of n, a(n) for n = 0..247</a>

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

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

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

%p a:= n-> (g-> coeff(series(1+(g@@(n+2))(x), x, n+1), x, n)*n!)(x-> exp(x)-1):

%p seq(a(n), n=0..20);

%p # second Maple program:

%p A:= proc(n, k) option remember; `if`(n=0 or k=0, 1,

%p       add(binomial(n-1, j-1)*A(j, k-1)*A(n-j, k), j=1..n))

%p     end:

%p a:= n-> A(n, n+1):

%p seq(a(n), n=0..20);

%p # third Maple program:

%p b:= proc(n, t, m) option remember; `if`(n=0, `if`(t=0, 1,

%p       b(m, t-1, 0)), m*b(n-1, t, m)+b(n-1, t, m+1))

%p     end:

%p a:= n-> b(n$2, 0):

%p seq(a(n), n=0..20);

%t 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 a[n_] := b[n, n, 0];

%t Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, Nov 18 2023, after 3rd Maple program *)

%Y First upper diagonal of A144150.

%Y Cf. A139383, A261280.

%K nonn

%O 0,3

%A _Alois P. Heinz_, Aug 04 2021