Is calculation a part or just a result of Mathematics?

Calculations can serve as part of a whole of a mathematical work and in rare instances are the mathematical work itself (some very complicated counterexamples in Banach space and $C^*$-algebra theory have been like this). For instance, a paper I just put up on arXiv had some pretty grueling calculations. I was defining a family of integral operators on a dense subset of $L^2(\mathbb{R})$ and wanted to establish that they are isometries. This required quite a bit of really unfortunate calculation but there was no reasonable way around it. Then by extension theorems, the operators could be lifted to all of $L^2(\mathbb{R})$. In general this is a pretty hard thing to do and some degree of calculation is needed. However many computations do not serve as a true mathematical work. Like most things in life, there isn't much black and white; it's all shades of gray.

I think the question is a little unclear unless we specify what exactly is meant by calculation.

Using Karatsuba algorithm to calculate the product of two natural numbers is not really a part of mathematics. Proving that it actually works is. Calculating the definite integral of a polynomial function by using the known antiderivatives of monomials is not mathematics, showing that the antiderivatives are what they are and that they can be used to calculate the integral is.

Generally, evaluating an expression once we have an algorithm for evaluating it is not mathematics, but developing an algorithm which does that is (whether it does so in general or just in some particular case). Just because we use computers to prove the four-colouring theorem doesn't mean that the computers are part of the mathematics. That's what I believe.

Of course, sometimes we have to actually apply some algorithm to prove something, but it doesn't mean that the actual calculation is a part of the mathematics. It is that no more than measuring length is a part of physics.