Reduction of the sum of the weight equal powers to explicit combinatorial representation

The paper contains the proof of the statement that the component of the sum of weighted powers with natural bases and equal parameters, dependent on weight coefficients, is equal to the sum of products of binomial and weight coefficients. It is proved also, that the component of this sum, independent of weight coefficients, is the algebraic sum of products of binomial coefficients and powers of natural numbers. Explicit combinatorial representation of the sum of the weighted equal powers contains the magnitudes taken from proved equalities.