is it true that ?
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Just thoughts.
L is left part, R is right part.
L = PL1 + PL2 + ... + PLk
R = PR1 + PR2 + ... + PRk
My theory is for all i = 1..k PLi > PRi.
Induction. Base i = 1.
So for x > 0: PL1 > PR1
And I really don't know what's next (if I'm on the right path).
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 ;)