Chandelier (noch nicht ├╝bersetzt)

Problem 768

A certain type of chandelier contains a circular ring of $n$ evenly spaced candleholders.
If only one candle is fitted, then the chandelier will be imbalanced. However, if a second identical candle is placed in the opposite candleholder (assuming $n$ is even) then perfect balance will be achieved and the chandelier will hang level.

Let $f(n,m)$ be the number of ways of arranging $m$ identical candles in distinct sockets of a chandelier with $n$ candleholders such that the chandelier is perfectly balanced.

For example, $f(4, 2) = 2$: assuming the chandelier's four candleholders are aligned with the compass points, the two valid arrangements are "North & South" and "East & West". Note that these are considered to be different arrangements even though they are related by rotation.

You are given that $f(12,4) = 15$ and $f(36, 6) = 876$.

Find $f(360, 20)$.