Products of Partitions

A partition of an integer is a way of representing a positive integer as a sum of other positive integers. For example, 4 can be partitioned in the following 5 ways

\[4=1+1+1+1=2+1+1=2+2=3+1\]

The product reduction of each partition is

\[4, \ 1\times1\times1\times1, \ 2\times1\times1, \ 2\times2, \ 3\times1\]

Out of these 5 products, \(2\) of the products are odd (\(3\times1\) and \(1\times1\times1\times1\)).

How many odd product reductions of the partitions of 40 are there?

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