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Computation of Euler’s type sums of the products of Bernoulli numbers. (English) Zbl 1228.11025
Summary: In this work, the authors present several formulas which compute the following Euler type and Dilcher type sums of the products of Bernoulli numbers $B\sb n$: $$\Omega\sp {(m)}\sb n := \sum\Sb j\sb 1+\dots+j\sb m=n\\ (j\sb 1,\dots,j\sb m\geq1)\endSb \left(\matrix2n\\ 2j\sb 1,\dots,2j\sb m\endmatrix\right)B\sb {2j\sb 1}\cdots B\sb {2j\sb m} $$ and $$ \Delta\sp {(m)}\sb n := \sum\Sb j\sb 1+\dots+j\sb m=n\\ (j\sb 1,\dots,j\sb m\geq0)\endSb\left(\matrix2n\\ 2j\sb 1,\dots,2j\sb m\endmatrix\right)B\sb {2j\sb 1}\cdots B\sb {2j\sb m} $$ respectively, where $$ \left(\matrix n\\ k\sb 1,\dots,k\sb m\endmatrix\right)={n!\over k\sb 1!\cdots k\sb m!} $$ denotes, as usual, the multinomial coefficient.

11B68Bernoulli and Euler numbers and polynomials
Full Text: DOI
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