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Let $\alpha \in (0,1)$ be a parameter. I would like to define a function $$ f_{\alpha}(n) := \binom{n}{\alpha n},$$ in order to later get a series expansion

Series[Subscript[f, α][n], {n, ∞, 0}]

as a function of $\alpha$.

How can I define such a function?

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    $\begingroup$ f[a_][n_] := Binomial[n, a n], and e.g. f[0.5][3.23] yields 3.85957; and then Series[f[a][n], {n, Infinity, 0}] returns something. $\endgroup$ – corey979 Nov 21 '19 at 15:28
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Try this:

f[α_Integer, n_Integer] := n!/(α!*(n - α)!)

Check:

f[2, 4]



   (*  6  *)

Have fun!

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