[seqfan] boxes in boxes

[Jon Wild]

Dear seqfans,

Say you have a bunch of boxes, each labelled by an integer. Boxes can go 
inside other boxes. The sequence I was just thinking about is the number 
of ways I can give you a collection of boxes whose labels sum to n. For 
n=3 for example, I can give you:

one box labelled 3;
one box labelled 2 and a separate box labelled 1;
a box labelled 2 with a box labelled 1 inside it;
a box labelled 1 with a box labelled 2 inside it;
three separate boxes labelled 1;
a box labelled 1 with another box labelled 1 inside it, plus a separate 
box labelled 1;
a box labelled 1 with two separate boxes labelled 1 both inside it;
a box labelled 1 with a box labelled 1 inside it, with a third box 
labelled 1 nested inside the second box.

That's 8 ways.

I checked by hand up to n=5 and got 1,3,8,24,71. Can someone confirm that 
this sequence is A052855, "Number of forests of rooted trees of nonempty 
sets with n points. (Each node is a set of 1 or more points.)"?

If so I'll add a comment, and if not I'll submit the sequence.

Jon