# Binomial Coefficients Summation $\sum_{k=0}^n a^k \binom{n}{k}^2$ [duplicate]

Let $$a>0.$$ Is there a closed form expression for the following summation: $$\sum_{k=0}^n a^k \binom{n}{k}^2$$?

• Well, according to Wolfram Alpha it is possible to express it in terms of Legendre polynomials. Do you consider this a closed form? Commented Jan 12, 2020 at 18:28
• Could you please post the expression in terms of Legendre polynomials if possible ? Thanks! Commented Jan 12, 2020 at 18:45
• For $a\neq 1$: $$\sum_{k=0}^n a^k \binom{n}{k}^2=(1-a)^n P_n\left(\frac{a+1}{1-a}\right)$$ For $a=1$: $$\sum_{k=0}^n \binom{n}{k}^2=\binom{2n}{n}$$ Commented Jan 12, 2020 at 18:48
• Some related older posts: Closed form for $S= \sum\limits_{k=0}^n x^k \binom{n}{k}^2$ and Evaluation of $\sum_{k=0}^n{n\choose k}^2u^k$. They can be found using Approach0 or SearchOnMath. See also: How to search on this site? Commented Jan 12, 2020 at 18:49
• Thanks a lot! it answers my question. Commented Jan 12, 2020 at 18:53