Geodesic
On the One Property of the Free Components Concerning to the Sum of Equal Powers
Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, no. 3 (2014), pp. 161-168.

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The given paper contains the proof of that the number of combinatorial arrangements coincides with free components of the sums of equal powers with the natural bases and parameters in the presence of the simple equality connecting elements of these arrangements. In the proof the modified exposition of the components participating in formation of the sum of equal powers is used. This exposition becomes simpler and led to an aspect of product of binomial factors. Other variants of construction of corresponding product of binomial factors do not exist here. The received proof allows both to represent number of arrangements in the form of product, and to apply at this representation summation elements. Thus, the number of arrangements supposes characteristic expression not only in the form of product of its elements.
Keywords: sum of equal powers, free components, number of arrangements, binomial factors. 
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A. I. Nikonov. On the One Property of the Free Components Concerning to the Sum of Equal Powers. Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, no. 3 (2014), pp. 161-168. http://geodesic.mathdoc.fr/item/VSGTU_2014_3_a12/

[1] Beery J., Sums of Powers of Positive Integers, Loci (July 2010), 2010 | DOI

[2] Oral H. K., Unal H., Extending al-Karaji's Work on Sums of Odd Powers of Integers, Loci (August 2011), 2011 | DOI

[3] Wang X., Yang S., “On solving equations of algebraic sum of equal powers”, Science in China Series A: Mathematics, 49:9 (2006), 1153–1157 | DOI | MR | Zbl

[4] De Koninck J.-M., Luca F., “Integers divisible by sums of powers of their prime factors”, Journal of Number Theory, 128:3 (2008), 557–563 | DOI | MR | Zbl

[5] Torabi-Dashti M., “Faulhaber's Triangle”, The College Mathematics Journal, 42:2 (2011), 96–97 | DOI | MR | Zbl

[6] Almismari N., A new method to express sums of power of integers as a polynomial equation, viXra:, 2012, 9 pp. 1211.0102

[7] Guo S., Shen Y., “On Sums of Powers of Odd Integers”, Journal of Mathematical Research with Applications, 33:6 (2013), 666–672 | DOI | MR | Zbl

[8] Suprijanto D., Rusliansyah, “Observation on sums of powers of integers divisible by four”, Applied Mathematical Sciences, 8:45 (2014), 2219–2226 | DOI | MR

[9] Cereceda J. L., “A determinant formula for sums of powers of integers”, International Mathematical Forum, 9:17 (2014), 785–795 | DOI

[10] 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 [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 2010, no. 5(21), 313–317 (In Russian) | DOI

[11] 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 [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 2012, no. 1(26), 223–232 (In Russian) | DOI

[12] Nikonov A. I., “Combinatorial representation of the sum of the weighted equal powers of members of an arithmetical progression”, Vestn. Samar. Gos. Tekhn. Univ. Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 2013, no. 4(33), 184–191 (In Russian) | DOI | Zbl

[13] Haggarty R., Discrete mathematics for computing, Addison-Wesley, Harlow, 2002, 235+xii pp. | Zbl

[14] Strang G., Linear Algebra and its Applications. 2nd ed., Academic Press, New York, San Francisco, London, 1980, 414+xi pp. | DOI | MR | Zbl

[15] Riordan J., An introduction to combinatorial analysis, Princeton University Press, Princeton, New Jersey, 1980, 244+xii pp. | MR | Zbl