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This is the formula that you get by expanding $$\sinh^k z = \left( e^z - e^{-z}\over 2\right)^k$$ by the binomial theorem. These numbers are essentially central factorial numbers.
A closed form is unlikely. For fixed $n$ (or $n$ small compared with $k$) the main contribution will come from the terms where for some $j$, $\gamma_j =k$ and $\gamma_i=0$ for $i\ne j$, so the sum will be asymptotic to $n\cdot k!$. More generally, asymptotic expansion can be obtained in this case by taking the terms where all but one $\gamma_i$ is a fixed integer.
