A290357 The eighth Euler transform of the sequence with g.f. 1+x.

1, 1, 8, 36, 204, 1002, 5244, 26328, 133476, 667335, 3331117, 16516607, 81607176, 401407499, 1967534543, 9609826869, 46788348316, 227114265649, 1099339308308, 5307155062783, 25556511343601, 122773840789344, 588473630650319, 2814565652799711, 13433897987956859

Offset: 0

Author

Alois P. Heinz, Jul 28 2017

Keywords

• nonn

Comments

Also the number of 8-level rooted trees with n leaves. All n leaves are in level 8.

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=8 of A290353.

Cf. A290356.

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, 8):
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, 8], {n, 0, 30}] (* Indranil Ghosh, Jul 30 2017, after Maple code *)

Formula

G.f.: Product_{j>0} 1/(1-x^j)^A290356(j).