What Pascal Identity Should I use?

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 \(b_{5,6} = 5.\)

Image Credit: Wikipedia Pascal Triangle

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