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This is a collection of definitions, notations and proofs for the Bernoullinumbers $B_n$ appearing in formulas for the sum of integer powers, some ofwhich can be found scattered in the large related historical literature inFrench, English and German. We provide elementary proofs for the originalconvention with ${\mathcal B}_1=1/2$ and also for the current convention with$B_1=-1/2$, using only the binomial theorem and the concise Blissard symbolic(umbral) notation.

Definitions, notations and proofs for Bernoulli numbers