# What Pascal Identity Should I use?

Number Theory Level 3

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 $$b_{5,6} = 5.$$

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