For $r \in \mathbb{N}$, let $a_r$ denote the number of ways to distribute $r$ identical objects into 3 identical boxes so that the box with the fewest objects has an odd number of objects. Find a generating function for $(a_r)$.
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Building on your work in the earlier question, you need to have an odd number of 3s in the conjugate partition. So that factor in the generating function is $$ x^3 + x^9 + x^{15} + \cdots = x^3(1 + x^6 + x^{12} + \cdots) = \frac{x^3}{1-x^6}$$ giving, altogether, $$ \sum_{n \ge 0} a(n) x^n = \prod_{i \ge 1} \frac{1}{1-x} \cdot \frac{1}{1-x^2} \cdot \frac{x^3}{1-x^6}. $$