Computation of Euler’s type sums of the products of Bernoulli numbers

In this work, the authors present several formulas which compute the following Euler’s type and Dilcher’s type sums of the products of Bernoulli numbers B n : Ω n ( m ) ≔ ∑ j 1 + ⋯ + j m = n ( j 1 , … , j m ≧ 1 ) 2 n 2 j 1 , … , 2 j m B 2 j 1 ⋯ B 2 j m and Δ n ( m ) ≔ ∑ j 1 + ⋯ + j m = n ( j 1 , … , j m ≧ 0 ) 2 n 2 j 1 , … , 2 j m B 2 j 1 ⋯ B 2 j m respectively, where n k 1 , … , k m = n ! k 1 ! ⋯ k m ! denotes, as usual, the multinomial coefficient.