Yeah, it is true. But proving this is a bit painful. Here is the only way I know:

Reminder: generating function of sequence is formal series .

Consider amount of ways to split number in sum of distinct numbers. Generating function for this sequence is . Now consider number of ways to split number in sum of not necessary distinct odd numbers. Generating function for this sequence is . Let's multiply each functions by . Then first one becomes and the second one become . Thus number of ways to split in distinct numbers equals number of ways to split in in odd numbers (you can also prove it with some combinatorics, I guess).

Now consider which is the number of ways to split in distinct numbers. . Thus if then . Then .

And with odd numbers part we have . Since , we have equation from the statement now.

Understanding of correctness of each transition in proof is left to reader as an excercise ;)