This paper presents the formula to calculate the sums of products of the Bernoulli numbers in the form \[ um_{ubstack{1eq k_1eq[\frac n2] 1eq k_2[\frac{n-k_1}2] \vdots 1eq k_meq[\frac{n-k_1-\dots-k_{m-1}}2]}} A_{k_1,k_2,\dots,k_m}B_{2k_1}\dots B_{2k_m}B_{2(n-k_1-\dots-k_m)}, \] where $A_{k_1,k_2,\dots,k_m}$ is a certain sequence of rational numbers.