I'll give it a shot.

Spoiler:The answer is .

I claim the following:

(1)when and zero otherwise.

(2)If we let , then

where ( = identity).

If these two claims hold, then

.

Proof of (1):The -th entry in is

.

The expression is 1 only when (zero otherwise) and is 1 only when and (zero otherwise). Therefore,(1)follows.

Proof of (2):Let . Then such that .

: is an odd number, is an odd number.Case 1

and , so

Clearly, , so from(1), .

: is odd, is even.Case 2

and , so

.

Since , , so from(1), .

: is even, is odd.Case 3

.

Since , , so .

is even, is even.Case 4:

.

If , then (same with and ).

If and , then let be the odd number between and and pick such that . Then is an odd number or an even number that's not , so from Cases 1 & 2, .

This finishes the proof.

Happy Holidays!