Inspired by Thaddeus

A composition of an integer is a way of representing a positive integer as a sum of other positive integers, where the order of the terms matter. For example, 3 can be composed in the following 4 ways:

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

The product reduction of a representation, refers to multiplying each of these terms together. For example, we would obtain the values of \( 3, 2, 2, 1 \) respectively. Out of these 4 products, 2 of them are odd and 2 of them are even.

How many odd product reductions of the composition of 10 are there?


Inspiration

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