Bernoulli number

\ [S_p (n) = \ Sigma_ {k = 0} ^ nk ^ p \\ G (z, n) = \ Sigma_ {p = 0} ^ {\ infty} Sp (n) / p! Z ^ p = \ Sigma_ {k = 0} ^ n \ Sigma_ {p = 0} ^ {\ infty} (kz) ^ p / p! \\ G (z, n) = \ Sigma_ {k = 0} ^ ne ^ {kz } = (e ^ {(n + 1) z} -1) / (e ^ z-1) \\ z / (e ^ z-1) = \ Sigma_ {k = 0} ^ {\ infty} B_k / k! * z ^ k \\ (e ^ {(n + 1) z} -1) / z = (n + 1) \ Sigma_ {k = 0} ^ {\ infty} ((n + 1) z) ^ k / (k + 1) / k! \\ G (z, n) = \ Sigma_ {k = 0} ^ {\ infty} z ^ k / k! (n + 1) \ Sigma_ {j = 0} ^ {k} C_k ^ jBj * (n + 1) ^ {kj} / (k-j + 1) \\ Sp (n) = (n + 1) \ Sigma_ {j = 0} ^ pC_p ^ jB_j * ( n + 1) ^ {pj} / (p-j + 1) \\ B_0 = 1, B_1 = -1 / 2, B_2 = 1/6, B_3 = 0, B_4 = -1 / 30 \\ S_2 (n ) = (n + 1) ^ 2 / 3- (n + 1) / 2 + 1/6 = (n + 1) ^ 3 / 3- (n + 1) ^ 2/2 + (n + 1) / 6 \\ S_1 (n) = (n + 1) ^ 2 / 2- (n + 1) / 2 = (n + 1) * n / 2 \]