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A290356 The seventh Euler transform of the sequence with g.f. 1+x.

%I A290356 #18 Aug 03 2017 17:40:34
%S A290356 1,1,7,28,140,602,2772,12166,54046,236093,1030101,4458247,19223202,
%T A290356 82448782,352247250,1498724840,6353940527,26844401919,113051495750,
%U A290356 474652297902,1987159118837,8296760311018,34551340915438,143533939056129,594877730354756
%N A290356 The seventh Euler transform of the sequence with g.f. 1+x.
%C A290356 Also the number of 7-level rooted trees with n leaves. All n leaves are in level 7.
%H A290356 Alois P. Heinz, <a href="/A290356/b290356.txt">Table of n, a(n) for n = 0..1000</a>
%H A290356 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 A290356 <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>
%F A290356 G.f.: Product_{j>0} 1/(1-x^j)^A290355(j).
%p A290356 with(numtheory):
%p A290356 b:= proc(n, k) option remember; `if`(n<2, 1, `if`(k=0, 0, add(
%p A290356       add(b(d, k-1)*d, d=divisors(j))*b(n-j, k), j=1..n)/n))
%p A290356     end:
%p A290356 a:= n-> b(n, 7):
%p A290356 seq(a(n), n=0..30);
%t A290356 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, 7], {n, 0, 30}] (* _Indranil Ghosh_, Jul 30 2017, after Maple code *)
%Y A290356 Column k=7 of A290353.
%Y A290356 Cf. A290355.
%K A290356 nonn
%O A290356 0,3
%A A290356 _Alois P. Heinz_, Jul 28 2017