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A290358 The ninth Euler transform of the sequence with g.f. 1+x.

%I A290358 #16 Aug 03 2017 17:42:32
%S A290358 1,1,9,45,285,1575,9237,52221,297633,1676364,9425128,52688379,
%T A290358 293582296,1629482947,9015732880,49727160669,273504111761,
%U A290358 1500271605182,8209029290412,44811239964075,244069307558722,1326536980923855,7195340066129605,38953817605037254
%N A290358 The ninth Euler transform of the sequence with g.f. 1+x.
%C A290358 Also the number of 9-level rooted trees with n leaves. All n leaves are in level 9.
%H A290358 Alois P. Heinz, <a href="/A290358/b290358.txt">Table of n, a(n) for n = 0..1000</a>
%H A290358 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 A290358 <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>
%F A290358 G.f.: Product_{j>0} 1/(1-x^j)^A290357(j).
%p A290358 with(numtheory):
%p A290358 b:= proc(n, k) option remember; `if`(n<2, 1, `if`(k=0, 0, add(
%p A290358       add(b(d, k-1)*d, d=divisors(j))*b(n-j, k), j=1..n)/n))
%p A290358     end:
%p A290358 a:= n-> b(n, 9):
%p A290358 seq(a(n), n=0..30);
%t A290358 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, 9], {n, 0, 30}] (* _Indranil Ghosh_, Jul 30 2017, after Maple code *)
%Y A290358 Column k=9 of A290353.
%Y A290358 Cf. A290357.
%K A290358 nonn
%O A290358 0,3
%A A290358 _Alois P. Heinz_, Jul 28 2017