[seqfan] decomposing partitions, is this new?

[Wouter Meeussen]

look at A000712(k) =partitions of k into parts of 2 kinds :

EXAMPLE:  Assume there are integers of two kinds k and k' then a(3) = 10 
since 3 has the following partitions into parts of two kinds: 111, 111', 
11'1', 1'1'1', 12, 1'2, 12', 1'2', 3, and 3'. [W. Edwin Clark, Jun 24 2011]

A000712 has a simple GF:  1/prod(m>=1, 1-x^m )^2

I found that the # of partitions of n can be decomposed into terms of 
A000712(k)  with values of k <= Floor[n/2]+1  as:

n=1; k={1},
n=2; k={2},
n=3; k={2, 1},
n=4; k={3},
n=5; k={3, 2},
...
n=33;k={17, 16, 10, 7},
n=34;k={18, 15, 13, 4},
n=35;k={18, 17, 11, 8},
n=36;k={19, 16, 14, 5, 1}

producing
p(1)=1,
p(2)=2,
p(3)=2 +1 ,
p(3)=5,
p(3)=5+ 2,
...
p(33)=5822+ 3956+ 300+ 65,
p(34)=8470+ 2665+ 1165+ 10,
p(35)=8470+ 5822+ 481+ 110,
p(36)=12230+ 3956+ 1770+ 20+ 1


The count of terms is surprisingly low: (for n=1..36)
{1, 1, 2, 1, 2, 2, 2, 2, 2, 3, 2, 3, 2, 3, 3, 3, 3, 3, 3, 3, 4, 3, 4, 3, 4, 
3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5}

Question:  is this somewhere implied in the descriptions of A000712 ?
(I don't see it)

Wouter.