0,3

Or, if the initial 0 is omitted, this is the binomial transform of the partition numbers p(1), p(2), ... = 1, 2, 3, 5, 7, 11, 15, 22, 30, ... (A000041 without the initial 1).

The most precise definition of this sequence is the Maple combstruct command given below. See the first Wieder link for further details.

Sequence appears to have a rational o.g.f. - Ralf Stephan, May 18 2007

Starting (1, 3, 8, 21, 54, 137,...), = row sums of triangle A137151 - Gary W. Adamson (qntmpkt(AT)yahoo.com), Jan 23 2008

N. J. A. Sloane and Thomas Wieder, The Number of Hierarchical Orderings, Order 21 (2004), 83-89.

Thomas Wieder, Expanded definitions of A103446 and A025168

Thomas Wieder, Home Page.

Thomas Wieder, (Old) Home Page.

Let {} denote a set, [] a list and Z an unlabeled element.

a(3) = 8 because we have {[[Z]],[[Z]],[[Z]]}, {[[Z],[Z]],[[Z]]}, {[[Z],[Z],[Z]]}, {[[Z],[Z,Z]]}, {[[Z,Z],[Z]]}, {[[Z,Z]],[[Z]]}, {[[Z]],[[Z,Z]]}, {[[Z,Z,Z]]}.

with(combstruct); SubSetSeqU := [T, {T=Subst(U, S), S=Set(U, card>=1), U=Sequence(Z, card>=1)}, unlabeled]; [seq(count(SubSetSeqU, size=n), n=0..30)];

allstructs(SubSetSeq, size=3); # to get the structures for n=3 - this output is shown in the example lines.

Cf. A025168, A034691, A050351.

Cf. A137151.

Sequence in context: A027930 A038200 A030015 this_sequence A094723 A127358 A077849

Adjacent sequences: A103443 A103444 A103445 this_sequence A103447 A103448 A103449

nonn

Thomas Wieder (wieder.thomas(AT)t-online.de), Feb 06 2005; revised Feb 20 2006

I can confirm that the terms shown are the binomial transform of the partition sequence 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, ... (A000041 without the a(0) term). - N. J. A. Sloane (njas(AT)research.att.com) - May 18 2007