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%I #38 Apr 08 2019 03:41:21
%S 1,1,5,15,55,170,571,1789,5727,17836,55627,171169,524879,1595896,
%T 4829894,14527981,43497312,129588391,384430264,1135607519,3341662498,
%U 9796626673,28620419254,83334382425,241879403752,699937499318,2019607806247,5811320364410,16677611788799
%N Number of 5-level rooted trees with n leaves.
%H Alois P. Heinz, <a href="/A007714/b007714.txt">Table of n, a(n) for n = 0..1000</a>
%H P. J. Cameron, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL3/groups.html">Sequences realized by oligomorphic permutation groups</a>, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
%H 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 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%H <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>
%F Euler transform applied 4 times to all-1's sequence.
%p with(numtheory): etr:= proc(p) local b; b:=proc(n) option remember; local d,j; if n=0 then 1 else add(add(d*p(d), d=divisors(j)) *b(n-j), j=1..n)/n fi end end: b[0]:= etr(1): for k from 1 to 2 do b[k]:= etr(b[k-1]) od: a:= etr(b[2]): seq(a(n), n=0..25); # _Alois P. Heinz_, Sep 08 2008
%t i[ n_, m_ ] := 1 /; m==1 || n==0; i[ n_, m_ ] := (i[ n, m ]=1/n Sum[ i[ k, m ] Plus @@ ((# i[ #, m-1 ])& /@ Divisors[ n-k ]), {k, 0, n-1} ]) /; n>0 && m>1
%t (* Second program: *)
%t A[0|1, _] = A[_, 1] = 1; A[n_, k_] := A[n, k] = Sum[DivisorSum[j, A[#, k-1] * #&]*A[n-j, k], {j, 1, n}]/n;
%t a[n_] := A[n, 5];
%t Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Jun 16 2018, after A290353 *)
%Y Column k=5 of A290353.
%K easy,nonn
%O 0,3
%A _N. J. A. Sloane_
%E More terms from _Christian G. Bower_, Aug 15 1998
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