Combination is odd or even I saw in other posts people asking how to know if a combination is odd or even, but in my case the formula for the combination is a little bit different:

$$C(n,k) = \frac{(n+k-1)!}{k!(n-1)!}$$

Can I check if the result is odd or even without calculate the value ?
I have the factorials pre-calculated with mod $10^9+7$, and if I use the values in the formula, results in a wrong answer.

Max value of N = 664579

Max value of K = 10000000
Actually what you are considering is $C(n+k-1,k)$ so you should just apply any formula you have for "usual combination" to this one.
Actually what you are considering is $C(n+k-1,k)$ so you should just apply any formula you have for "usual combination" to this one.