A290355 The sixth Euler transform of the sequence with g.f. 1+x.

1, 1, 6, 21, 91, 336, 1337, 5026, 19193, 71769, 268272, 992676, 3659116, 13400426, 48863017, 177299790, 640713627, 2305930966, 8268556438, 29544196129, 105215495691, 373523546056, 1322096328899, 4666327388034, 16425341129078, 57667752483279, 201967215942032

Offset: 0

Author

Alois P. Heinz, Jul 28 2017

Keywords

• nonn

Comments

Also the number of 6-level rooted trees with n leaves. All n leaves are in level 6. a(2) = 6:

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Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000
• 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

Column k=6 of A290353.

Cf. A007714.

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, 6):
seq(a(n), n=0..30);

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, 6], {n, 0, 30}] (* Indranil Ghosh, Jul 30 2017, after Maple code *)

Formula

G.f.: Product_{j>0} 1/(1-x^j)^A007714(j).