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

%I A290355 #21 Aug 03 2017 17:41:02
%S A290355 1,1,6,21,91,336,1337,5026,19193,71769,268272,992676,3659116,13400426,
%T A290355 48863017,177299790,640713627,2305930966,8268556438,29544196129,
%U A290355 105215495691,373523546056,1322096328899,4666327388034,16425341129078,57667752483279,201967215942032
%N A290355 The sixth Euler transform of the sequence with g.f. 1+x.
%C A290355 Also the number of 6-level rooted trees with n leaves. All n leaves are in level 6. a(2) = 6:
%C A290355 :    o      o      o      o      o      o
%C A290355 :    |      |      |      |      |     ( )
%C A290355 :    o      o      o      o      o     o o
%C A290355 :    |      |      |      |     ( )    | |
%C A290355 :    o      o      o      o     o o    o o
%C A290355 :    |      |      |     ( )    | |    | |
%C A290355 :    o      o      o     o o    o o    o o
%C A290355 :    |      |     ( )    | |    | |    | |
%C A290355 :    o      o     o o    o o    o o    o o
%C A290355 :    |     ( )    | |    | |    | |    | |
%C A290355 :    o     o o    o o    o o    o o    o o
%C A290355 :   ( )    | |    | |    | |    | |    | |
%C A290355 :   o o    o o    o o    o o    o o    o o
%H A290355 Alois P. Heinz, <a href="/A290355/b290355.txt">Table of n, a(n) for n = 0..1000</a>
%H A290355 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 A290355 <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>
%F A290355 G.f.: Product_{j>0} 1/(1-x^j)^A007714(j).
%p A290355 with(numtheory):
%p A290355 b:= proc(n, k) option remember; `if`(n<2, 1, `if`(k=0, 0, add(
%p A290355       add(b(d, k-1)*d, d=divisors(j))*b(n-j, k), j=1..n)/n))
%p A290355     end:
%p A290355 a:= n-> b(n, 6):
%p A290355 seq(a(n), n=0..30);
%t A290355 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 *)
%Y A290355 Column k=6 of A290353.
%Y A290355 Cf. A007714.
%K A290355 nonn
%O A290355 0,3
%A A290355 _Alois P. Heinz_, Jul 28 2017