Let us assume that $X \geq 0$ is a nonnegative integer valued random variable with the below mass function.
$\mathbb {P} (X = k) = \frac {2^k \, {C \choose k} \, {C - k \choose \frac {M - k} {2}}} {{2C \choose M}} \quad \left\{ \begin {array} {ll} \textrm {$k$ is odd} & \textrm {if $M$ is odd} \\ \textrm {$k$ is even} & \textrm {if $M$ is even} \end {array} \right.$
$C$ is a known constant, and $M \in [1, \, C]$ is also a known constant. The expectation of $X$ is known: $\mathbb {E} X = \frac {M (2C - M)} {C (2C - 1)}$.
I would like to calculate the $\mathbb {E} \left( \frac {1} {X} \right| \left. X \neq 0 \right)$ conditional expectation.
$\mathbb {E} \left( \frac {1} {X} | X \neq 0 \right) = \frac {1} {\left( 1 - \mathbb {P} (X = 0) \right) \, {2C \choose M}} \, \sum_{k = 1}^{C} \frac {2^k} {k} \, {C \choose k} \, {C - k \choose \frac {M - k} {2}} \quad \left\{ \begin {array} {ll} \textrm {$k$ is odd} & \textrm {if $M$ is odd} \\ \textrm {$k$ is even} & \textrm {if $M$ is even} \end {array} \right.$
I would like to ask for your kind help for this summation. Approximations are also helpful and sufficient.