An expression like newton's binomial

Question

I have the following sum, it looks just like Newton's Binomial the only difference is that it's twice the sum,

I have $$ \sum_0^{n} \left( \begin{matrix} 2n \\ 2k \end{matrix} \right) a ^{2k} b^{2n - 2k}$$ and I want to express this with $(a+b)^n$ or something in that sort. Is there a way to do it?

how do I turn this into Newton's binomial, it's so close but so far away. Is there some sort of manipulation im missing here?

Answer 1

Using the binomial theorem to expand $$ \frac12\left((a+b)^{2n} + (a-b)^{2n}\right) $$ and then simplify terms of equal degree in $a$ and $b$ will give you the sum you seek.