A007714 Number of 5-level rooted trees with n leaves.

1, 1, 5, 15, 55, 170, 571, 1789, 5727, 17836, 55627, 171169, 524879, 1595896, 4829894, 14527981, 43497312, 129588391, 384430264, 1135607519, 3341662498, 9796626673, 28620419254, 83334382425, 241879403752, 699937499318, 2019607806247, 5811320364410, 16677611788799

Offset: 0

Author

N. J. A. Sloane

Keywords

• easy
• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000
• P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
• B. A. Huberman and T. Hogg, Complexity and adaptation, Evolution, games and learning (Los Alamos, N.M., 1985). Phys. D 22 (1986), no. 1-3, 376-384.
• N. J. A. Sloane, Transforms
• Index entries for sequences related to rooted trees

Crossrefs

Column k=5 of A290353.

Programs

Maple

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

Mathematica

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
(* Second program: *)
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;
a[n_] := A[n, 5];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Jun 16 2018, after A290353 *)

Formula

Euler transform applied 4 times to all-1's sequence.

Extensions

More terms from Christian G. Bower, Aug 15 1998