Geodesic
Reduction of the sum of the weight equal powers to explicit combinatorial representation
Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, Tome 128 (2012) no. 3, pp. 163-169

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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.
Keywords: sum of weighted equal powers, combinatorial representation, weight coefficients.
Mots-clés : binomial coefficients 
A. I. Nikonov. Reduction of the sum of the weight equal powers to explicit combinatorial representation. Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, Tome 128 (2012) no. 3, pp. 163-169. http://geodesic.mathdoc.fr/item/VSGTU_2012_128_3_a15/
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[1] Nikonov A. I., “Converting the Sum of Weighted Degrees of Natural Numbers with the Same Parameters”, Vestn. Samar. Gos. Tekhn. Univ. Ser. Fiz.-Mat. Nauki, 2010, no. 1(20), 258–262 | DOI

[2] Nikonov A. I., “On One Property of the Weighed Sums of Equal Powers as Matrix Products”, Vestn. Samar. Gos. Tekhn. Univ. Ser. Fiz.-Mat. Nauki, 2010, no. 5(21), 313–317 | DOI

[3] Nikonov A. I., “The update exposition of the components organising the sum of weighted equal powers”, Vestn. Samar. Gos. Tekhn. Univ. Ser. Fiz.-Mat. Nauki, 2012, no. 1(26), 223–232 | DOI

[4] Vilenkin N. Ya., Combinatorics, Nauka, Moscow, 1969, 328 pp. | MR | Zbl