What Pascal Identity Should I use?

Let bk,nb_{k,n} be the kthk^\text{th} element of the nthn^\text{th} row of Pascal's triangle, with row 11 being {1},\{1\}, row 22 being {1,1},\{1,1\}, etc. Also, the 1st1^\text{st} element of each row is 1.1.

Find the value of

(b3,2015)2i=12013i3.(b_{3,2015})^2 - \sum_{i=1}^{2013} i^3.

Details and Assumptions:

  • For example, b1,1=1,b2,3=2,b_{1,1} = 1, b_{2,3} = 2, and b5,6=5.b_{5,6} = 5.

Image Credit: Wikipedia Pascal Triangle

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