[seqfan] a partition sequence

[David Newman]

Hi,

     I'm looking for someone to confirm and extend the following.

How many partitions of n are there such that the partition contains a
sub-multiset the sum of whose elements is k for all k<n?
An example:

for 6 there are 5 such partitions:

{3,2,1}      which has subsets  {1}, {2},{3},{1,3},{2,3},{1,2,3} with the
sums 1,2,3,4,5,6.
{3,1,1,1}   which has subsets {1} ,{1,1},{3},{3,1},{3,1,1},{3,1,1,1}  with
the sums 1,2,3,4,5,6
{2,2,1,1}   ...
{2,1,1,1,1} ...
{1,1,1,1,1,1}...

but other  partitions of 6 like {6}, {5,1}, {4,2},{3,3},{2,2,2} will not be
counted, because in each such case there is at least 1 number less than 6
which is not the sum of any subset of the given partition.

The numbers that I've gotten are a(1)=1, a(2)=1, a(3)=2, a(4)=2, a(5)=4,
a(6)=5, a(7)=8, a(8)=10, a(9)=12, a(10)=16

I've only done the computations by hand thus far, and the sequence of
numbers that I've gotten does not seem to be in the OEIS.