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If not, why not?

Also I don't mean Bernoulli number that are negative such as $B_4 = \frac{-1}{30}$ but numbers like $B_{-4} = ?$

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A very natural generalization of the Bernoulli-numbers are the zeta-values at negative arguments (scaled by the index). Since the zeta-function is defined for all complex numbers (except the numer 1) you can assume the zetas at positive integer arguments as "negative Bernoulli numbers". – Gottfried Helms Sep 20 '13 at 22:33

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