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
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
KEYWORD
easy,nonn
AUTHOR
STATUS
approved