figuring out an integer function $f(1) = 1\\
f(2) = 2\\
f(3) = 6\\
f(4) = 20\\
f(5) = 70\\
f(6) = 252\\
f(7) = 924\\
f(8) = 3432\\
f(9) = 12870$
Then what is $f(n)$ (where $n > 0$)?
I though about many many possibilities but still cannot figure out the expression.

What about this?
$$\frac{2^{n-1} (2 n-3)\text{!!}}{(n-1)!}$$
which gives
$$
\begin{array}{ccl}
 f(1) & = & 1 \\
 f(2) & = & 2 \\
 f(3) & = & 6 \\
 f(4) & = & 20 \\
 f(5) & = & 70 \\
 f(6) & = & 252 \\
 f(7) & = & 924 \\
 f(8) & = & 3432 \\
 f(9) & = & 12870 \\
 f(10) & = & 48620 \\
 f(11) & = & 184756 \\
 f(12) & = & 705432 \\
 f(13) & = & 2704156 \\
 f(14) & = & 10400600 \\
 f(15) & = & 40116600 \\
 f(16) & = & 155117520 \\
 f(17) & = & 601080390 \\
 f(18) & = & 2333606220 \\
 f(19) & = & 9075135300 \\
 f(20) & = & 35345263800 \\
\end{array}
$$
