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

%I A290357 #17 Aug 03 2017 17:41:58
%S A290357 1,1,8,36,204,1002,5244,26328,133476,667335,3331117,16516607,81607176,
%T A290357 401407499,1967534543,9609826869,46788348316,227114265649,
%U A290357 1099339308308,5307155062783,25556511343601,122773840789344,588473630650319,2814565652799711,13433897987956859
%N A290357 The eighth Euler transform of the sequence with g.f. 1+x.
%C A290357 Also the number of 8-level rooted trees with n leaves. All n leaves are in level 8.
%H A290357 Alois P. Heinz, <a href="/A290357/b290357.txt">Table of n, a(n) for n = 0..1000</a>
%H A290357 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 A290357 <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>
%F A290357 G.f.: Product_{j>0} 1/(1-x^j)^A290356(j).
%p A290357 with(numtheory):
%p A290357 b:= proc(n, k) option remember; `if`(n<2, 1, `if`(k=0, 0, add(
%p A290357       add(b(d, k-1)*d, d=divisors(j))*b(n-j, k), j=1..n)/n))
%p A290357     end:
%p A290357 a:= n-> b(n, 8):
%p A290357 seq(a(n), n=0..30);
%t A290357 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 *)
%Y A290357 Column k=8 of A290353.
%Y A290357 Cf. A290356.
%K A290357 nonn
%O A290357 0,3
%A A290357 _Alois P. Heinz_, Jul 28 2017