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

1, 1, 2, 6, 30, 170, 1337, 12166, 133476, 1676364, 23970089, 383172262, 6783362586, 131697494825, 2783238819896, 63605879539200, 1563127601683456, 41107799958703376, 1151957989511106438, 34268629198432285436, 1078577860182473404134, 35809701458658690462644

Offset: 0

Author

Alois P. Heinz, Jul 28 2017

Keywords

• nonn

Comments

a(n) is also the number of unlabeled rooted trees with exactly n leaves, all in level n. a(3) = 6:

: o o o o o o

: | | | / \ / \ /|\

: o o o o o o o o o o

: | / \ /|\ | | ( ) | | | |

: o o o o o o o o o o o o o o

: /|\ ( ) | | | | ( ) | | | | | | |

: o o o o o o o o o o o o o o o o o o

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..414
• B. A. Huberman and T. Hogg, Complexity and adaptation, Evolution, games and learning (Los Alamos, N.M., 1985). Phys. D 22 (1986), no. 1-3, 376-384.
• Index entries for sequences related to rooted trees

Crossrefs

Main diagonal of A290353.

Cf. A139383, A305725.

Programs

Maple

with(numtheory):
b:= proc(n, k) option remember; `if`(n<2, 1, `if`(k=0, 0, add(
      add(b(d, k-1)*d, d=divisors(j))*b(n-j, k), j=1..n)/n))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..25);

Mathematica

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 *)

Formula

a(n) = A290353(n,n).

Conjecture: a(n) ~ c * 2^n * n^(n-4/3) / Pi^n, where c = 4.4923... - Vaclav Kotesovec, Aug 14 2017