By examining the connection coefficients, we systematically review and extend (with an extra integer parameter) several double sum expressions for the Bernoulli numbers. New summation formulae are also established explicitly.

Carlitz' multiple sums involving cyclically symmetric products of binomial coefficients are extended by introducing weight monomials of $m$-variables. By making use of recursive reductions and the Lagrange expansion formula, we determine, in closed form, the related rational generating functions that significantly generalize the classical result discovered by Carlitz in 1965. Several novel applications are presented as consequences.

By means of the recursive construction method, we examine a class of multiple sums (with a free variable " $x$") for circular products of binomial coefficients. They are first expressed as coefficients of bivariate rational functions. Then the rational generating functions are determined by deftly employing Knuth's bracket calculus, resultants of polynomials, and Hadamard products of rational functions.