  In this paper, we show that the difference between the number of parts in the
odd partitions of $n$ and the number of parts in the distinct partitions of $n$
satisfies Euler's recurrence relation for the partition function $p(n)$ when
$n$ is odd. A decomposition of this difference in terms of the total number of
parts in all the partitions of $n$ is also derived. In this context, we
conjecture that for $k>0$, the series
  $$
  (q^2;q^2)_\infty \sum_{n=k}^\infty \frac{q^{{k\choose 2}+(k+1)n}}{(q;q)_n}
  \begin{bmatrix}
  n-1\\k-1
  \end{bmatrix}
  $$ has non-negative coefficients.
