Combinatorial representation of the sum of the weighted equal powers of members of an arithmetical progression

The correctness of equality which gives the combinatorial expression for the sum of the weighted equal powers of members of an arithmetical progression is found out. Such aspect provides usage of double summation of certain algebraic combinations with free and weight components of the given sum. Thus specified algebraic combinations also include binomial coefficients. Determination of required equality was made with use of comparison of real and hypothetical values.