Let $b_{k,n}$ be the $k^\text{th}$ element of the $n^\text{th}$ row of Pascal's triangle, with row $1$ being $\{1\},$ row $2$ being $\{1,1\},$ etc. Also, the $1^\text{st}$ element of each row is $1.$

Find the value of

$(b_{3,2015})^2 - \sum_{i=1}^{2013} i^3.$

**Details and Assumptions:**

- For example, $b_{1,1} = 1, b_{2,3} = 2,$ and $b_{5,6} = 5.$