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A007713 Number of 4-level rooted trees with n leaves. 16
1, 1, 4, 10, 30, 75, 206, 518, 1344, 3357, 8429, 20759, 51044, 123973, 299848, 719197, 1716563, 4070800, 9607797, 22555988, 52718749, 122655485, 284207304, 655894527, 1508046031, 3454808143, 7887768997, 17949709753, 40719611684, 92096461012, 207697731344 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

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

FORMULA

Euler transform applied thrice to all-1's sequence.

EXAMPLE

From Gus Wiseman, Oct 11 2018: (Start)

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:

((1)) ((2)) ((3)) ((4))

((11)) ((12)) ((13))

((1)(1)) ((111)) ((22))

((1))((1)) ((1)(2)) ((112))

((1)(11)) ((1111))

((1))((2)) ((1)(3))

((1))((11)) ((2)(2))

((1)(1)(1)) ((1)(12))

((1))((1)(1)) ((2)(11))

((1))((1))((1)) ((1)(111))

((11)(11))

((1))((3))

((2))((2))

((1))((12))

((1)(1)(2))

((2))((11))

((1))((111))

((1)(1)(11))

((11))((11))

((1))((1)(2))

((2))((1)(1))

((1))((1)(11))

((1)(1)(1)(1))

((11))((1)(1))

((1))((1))((2))

((1))((1))((11))

((1))((1)(1)(1))

((1)(1))((1)(1))

((1))((1))((1)(1))

((1))((1))((1))((1))

(End)

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: b0:= etr(1): b1:= etr(b0): a:= etr(b1): seq(a(n), n=0..30); # 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

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 *)

CROSSREFS

Column k=4 of A290353.

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

Sequence in context: A002220 A222807 A090578 * A058488 A036674 A006357

Adjacent sequences: A007710 A007711 A007712 * A007714 A007715 A007716

KEYWORD

easy,nonn

AUTHOR

N. J. A. Sloane.

STATUS

approved

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Last modified December 5 17:16 EST 2022. Contains 358588 sequences. (Running on oeis4.)