If the sum of the series $1^2 + 2 \cdot 2^2 + 3^2 + 2 \cdot 4^2 + 5^2 + \dots + 2 \cdot (n-1)^2 + n^2$ (when $n$ is odd) or $1^2 + 2 \cdot 2^2 + 3^2 + 2 \cdot 4^2 + \dots + 2 \cdot n^2$ (when $n$ is even) is given by $S_n = \frac{n(n+1)^2}{2}$ for even $n$,find the sum of the series when $n$ is odd.

• A
  $n^2(n + 1)$
• B
  $\frac{n^2(n - 1)}{2}$
• C
  $\frac{n^2(n + 1)}{2}$
• D
  $n^2(n - 1)$