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This identity can be written in the following form $$\sum_{i=0}^k (-1)^{k-i+1} B_i\cdot\binom{k+1}{i}=(-1)^{k+1}(k+1).$$ The latter is equivalent to $B_{k+1}(0)-B_{k+1}(-1)=(k+1)(-1)^k$ which is one of the basic properties of Bernoully polynomials.