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A290354 a(n) is the n-th term of the n-th Euler transform of the sequence with g.f. 1+x.

%I A290354 #26 Jun 21 2018 18:22:38
%S A290354 1,1,2,6,30,170,1337,12166,133476,1676364,23970089,383172262,
%T A290354 6783362586,131697494825,2783238819896,63605879539200,
%U A290354 1563127601683456,41107799958703376,1151957989511106438,34268629198432285436,1078577860182473404134,35809701458658690462644
%N A290354 a(n) is the n-th term of the n-th Euler transform of the sequence with g.f. 1+x.
%C A290354 a(n) is also the number of unlabeled rooted trees with exactly n leaves, all in level n. a(3) = 6:
%C A290354 :    o        o       o        o        o       o
%C A290354 :    |        |       |       / \      / \     /|\
%C A290354 :    o        o       o      o   o    o   o   o o o
%C A290354 :    |       / \     /|\     |   |   ( )  |   | | |
%C A290354 :    o      o   o   o o o    o   o   o o  o   o o o
%C A290354 :   /|\    ( )  |   | | |   ( )  |   | |  |   | | |
%C A290354 :  o o o   o o  o   o o o   o o  o   o o  o   o o o
%H A290354 Alois P. Heinz, <a href="/A290354/b290354.txt">Table of n, a(n) for n = 0..414</a>
%H A290354 B. A. Huberman and T. Hogg, <a href="https://doi.org/10.1016/0167-2789(86)90308-1">Complexity and adaptation</a>, Evolution, games and learning (Los Alamos, N.M., 1985). Phys. D 22 (1986), no. 1-3, 376-384.
%H A290354 <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>
%F A290354 a(n) = A290353(n,n).
%F A290354 Conjecture: a(n) ~ c * 2^n * n^(n-4/3) / Pi^n, where c = 4.4923... - _Vaclav Kotesovec_, Aug 14 2017
%p A290354 with(numtheory):
%p A290354 b:= proc(n, k) option remember; `if`(n<2, 1, `if`(k=0, 0, add(
%p A290354       add(b(d, k-1)*d, d=divisors(j))*b(n-j, k), j=1..n)/n))
%p A290354     end:
%p A290354 a:= n-> b(n$2):
%p A290354 seq(a(n), n=0..25);
%t A290354 b[n_, k_]:=b[n, k]=If[n<2, 1, If[k==0, 0, Sum[Sum[b[d, k - 1]*d, {d, Divisors[j]}] b[n - j, k], {j, n}]/n]]; Table[b[n, n], {n, 0, 30}] (* _Indranil Ghosh_, Jul 30 2017, after Maple code *)
%Y A290354 Main diagonal of A290353.
%Y A290354 Cf. A139383, A305725.
%K A290354 nonn
%O A290354 0,3
%A A290354 _Alois P. Heinz_, Jul 28 2017