A007713 - OEIS

%I #40 Feb 18 2023 14:45:11

%S 1,1,4,10,30,75,206,518,1344,3357,8429,20759,51044,123973,299848,

%T 719197,1716563,4070800,9607797,22555988,52718749,122655485,284207304,

%U 655894527,1508046031,3454808143,7887768997,17949709753,40719611684,92096461012,207697731344

%N Number of 4-level rooted trees with n leaves.

%H Alois P. Heinz, <a href="/A007713/b007713.txt">Table of n, a(n) for n = 0..1000</a>

%H P. J. Cameron, <a href="http://www.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 thrice to all-1's sequence.

%e From _Gus Wiseman_, Oct 11 2018: (Start)

%e Also the number of multiset partitions of multiset partitions of integer partitions of n. For example, the a(1) = 1 through a(4) = 30 multiset partitions are:

%e   ((1))  ((2))       ((3))            ((4))

%e          ((11))      ((12))           ((13))

%e          ((1)(1))    ((111))          ((22))

%e          ((1))((1))  ((1)(2))         ((112))

%e                      ((1)(11))        ((1111))

%e                      ((1))((2))       ((1)(3))

%e                      ((1))((11))      ((2)(2))

%e                      ((1)(1)(1))      ((1)(12))

%e                      ((1))((1)(1))    ((2)(11))

%e                      ((1))((1))((1))  ((1)(111))

%e                                       ((11)(11))

%e                                       ((1))((3))

%e                                       ((2))((2))

%e                                       ((1))((12))

%e                                       ((1)(1)(2))

%e                                       ((2))((11))

%e                                       ((1))((111))

%e                                       ((1)(1)(11))

%e                                       ((11))((11))

%e                                       ((1))((1)(2))

%e                                       ((2))((1)(1))

%e                                       ((1))((1)(11))

%e                                       ((1)(1)(1)(1))

%e                                       ((11))((1)(1))

%e                                       ((1))((1))((2))

%e                                       ((1))((1))((11))

%e                                       ((1))((1)(1)(1))

%e                                       ((1)(1))((1)(1))

%e                                       ((1))((1))((1)(1))

%e                                       ((1))((1))((1))((1))

%e (End)

%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: b0:= etr(1): b1:= etr(b0): a:= etr(b1): seq(a(n), n=0..30); # _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 etr[p_] := Module[{b}, b[n_] := b[n] = If[n == 0, 1, Sum[Sum[d*p[d], {d, Divisors[ j]}]*b[n-j], {j, 1, n}]/n]; b]; b0 = etr[Function[1]]; b1 = etr[b0]; a = etr[b1]; Table[a[n], {n, 1, 30}] (* _Jean-François Alcover_, Mar 05 2015, after _Alois P. Heinz_ *)

%Y Column k=4 of A290353.

%Y Main diagonal of A055886.

%Y Cf. A001970, A047968, A050342, A089259, A141268, A258466, A261049, A319066, A320328, A320330, A320331.

%K easy,nonn

%O 0,3

%A _N. J. A. Sloane_.