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The following function popped in my research: $$f(x)=\sum_{\array{0\le t\le k \\ t\equiv_p a}}{x \choose t}{n-x \choose k-t}$$

Where:

  1. n,k are natural numbers and $k\le n$.
  2. t is taken over all integers between $0$ and $k$ such that $t$ is equivalent to $a$ modulo a prime p.
  3. x is a natural number between 0 and n.

So the function is determined by the parameters $n,k,p,a$. I'm particularly interested in the case $p=3$ and $a=0$ (but also $a=1$ is relevant).

My main question is - is there a way to characterize (instead of simply computing) for which values of $x$ (given $n,k,p,a$) the values of $f(x)$ will be odd numbers?

The first step seems to be answering the question of when ${\alpha \choose \beta}$ is odd and when it is even. This question was asked here and received a beautiful solution; however, I fail to see a way in which to apply that solution to "my" function.

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