compositions of n into even parts

Question

I have found here {https://math.stackexchange.com/questions/2167885/compositions-of-n-into-odd-parts} that the number of integer compositions of n into k odd parts would be ${\frac{n+k-1}{2} \choose k-1}$.

I would like to find the number of integer compositions of n into k even parts. My guess is that it would be the same, but I do not see how to prove it.

Answer 1

Each positive, even number is at least equal to two, so you can subtract one to get an odd, but still positive, number.

Therefore, the number of ways to decompose $n$ into $k$ even numbers is the same as the number of compositions of $n-k$ into $k$ odd numbers; and for that you already have a formula.