[seqfan] Counting hexagonal partitions of n

[Allan Wechsler]

By "hexagonal partition" I mean an ordered partitions whose parts start at
some value, then increase by one, up to some maximum, then stay constant at
that maximum, for some number of steps, then decrease to an ending value.

For n=9, for example, the hexagonal partitions are

1+1+1+1+1+1+1+1+1
1+2+2+2+2
1+2+3+2+1
1+2+3+3
2+2+2+2+1
2+3+4
3+3+2+1
3+3+3
4+5
5+4
9

Unless I have missed some, there are 11 possibilities, so a(9) = 11.

I have collected some values in which I have fairly little confidence, but
I think that the number of such partitions of n is not in OEIS.

There is an intimate connection to A116513. Each hexagon that A116513
counts has at least one and as many as six "readings" as a hexagonal
partition.

Can someone provide some values with more confidence than I have been able
to do?