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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 ${B}_{n}$:

${{\Omega }}_{n}^{\left(m\right)}:=\sum _{\begin{array}{c}{j}_{1}+\cdots +{j}_{m}=n\\ \left({j}_{1},\cdots ,{j}_{m}\ge 1\right)\end{array}}\left(\begin{array}{c}2n\\ 2{j}_{1},\cdots ,2{j}_{m}\end{array}\right){B}_{2{j}_{1}}\cdots {B}_{2{j}_{m}}$

and

${{\Delta }}_{n}^{\left(m\right)}:=\sum _{\begin{array}{c}{j}_{1}+\cdots +{j}_{m}=n\\ \left({j}_{1},\cdots ,{j}_{m}\ge 0\right)\end{array}}\left(\begin{array}{c}2n\\ 2{j}_{1},\cdots ,2{j}_{m}\end{array}\right){B}_{2{j}_{1}}\cdots {B}_{2{j}_{m}}$

respectively, where

$\left(\begin{array}{c}n\\ {k}_{1},\cdots ,{k}_{m}\end{array}\right)=\frac{n!}{{k}_{1}!\cdots {k}_{m}!}$

denotes, as usual, the multinomial coefficient.

##### MSC:
 11B68 Bernoulli and Euler numbers and polynomials
OEIS
##### References:
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