You can represent a partition $k_1+\dots +k_m=n$, $k_1\geq\dots\geq k_m$ by its Young diagram: a row of $k_1$ squares, below it a row of $k_2$ squares ... and finally a row of $k_m$ squares.
A partition to different odd numbers $l_1>\dots >l_p$ can also be represented by a Young diagram: a hook of $l_1$ squares (if $l_1=2q_1+1$, the hook has one square in the corner and $q_1$ squares horizontally and $q_1$ squares vertically); then another hook of $l_2$ squares, etc. The diagrams obtained in this way are invariant wrt. the diagonal reflection.
The diagrams which are not invariant wrt. the diagonal refection come in pairs (a diagram and it reflection), so the number of all diagrams (with $n$ squares) is $2\times$something plus the number of invariant diagrams.
edit: a picture will say more: