### Realization of a Method for Calculating Bell Polynomials Based on Compositae of Generating Functions

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A. Bayad, Y. Simsek and H.M. Srivastava, Some array type polynomials associated with special numbers and polynomials, Appl. Math. Comput. 244 (2014), 149 – 157.

E.T. Bell, Partition polynomials, Ann. of Math. (2) 29(1/4) (1927-1928), 38 – 46.

D. Birmajer, J.B. Gil and M.D. Weiner, Colored partitions of a convex polygon by noncrossing diagonals, Discrete Math. 340(4) (2017), 563 – 571.

V.P. Bubnov, A.S. Eremin, N.A. Kovrizhnykh and I.V. Olemskoy, Comparative study of the advantages of structural numerical integration methods for ordinary differential equations, SPIIRAS Proceedings 4(53) (2017), 51 – 72.

L. Comtet, Advanced Combinatorics, D. Reidel Publishing Company (1974).

C. Gilson, F. Lambert, J. Nimmo and R. Willox, On the combinatorics of the Hirota D-operators, Proc. R. Soc. Lond. A 452(1945) (1996), 223 – 234.

Y. Jiang, B. Tian, W.-J. Liu, M. Li, P. Wang and K. Sun, Solitons, Backlund transformation, and Lax pair for the (2Å1)-dimensional Boiti-Leon-Pempinelli equation for the water waves, J. Math. Phys. 51(9) (2010), 1 – 11.

D.S. Kim and T. Kim, Some identities of Bell polynomials, Sci. China Math. 58(10) (2015), 1 – 10.

U. Kotta and M. Tonso, NLControl – a Mathematica package for nonlinear control systems, IFACPapersOnLine 50(1) (2017), 681 – 686.

M.I. Krivoruchenko, Trace identities for skew-symmetric matrices, Mathematics and Computer Science 1(2) (2016), 21 – 28.

V. Kruchinin, Derivation of Bell polynomials of the second kind, arXiv:1104.5065v1 (2011), 1 – 15.

D.V. Kruchinin and V.V. Kruchinin, A method for obtaining generating functions for central coefficients of triangles, J. Integer Seq. 15 (2012), 1 – 10.

D.V. Kruchinin and V.V. Kruchinin, Application of a composition of generating functions for obtaining explicit formulas of polynomials, J. Math. Anal. Appl. 404(1) (2013), 161 – 171.

D.V. Kruchinin and V.V. Kruchinin, Explicit formulas for some generalized polynomials, Appl. Math. Inf. Sci. 7(5) (2013), 2083 – 2088.

V.V. Kruchinin and D.V. Kruchinin, Composita and its properties, Journal of Analysis and Number Theory 2 (2014), 37 – 44.

D.V. Kruchinin, Y.V. Shablya, O.O. Evsutin and A.A. Shelupanov, Integer properties of a composition of exponential generating functions, in: Proc. Intl. Conf. of Numerical Analysis and Applied Mathematics, T. Simos and C. Tsitouras (eds.), American Institute of Physics Inc., 2017, pp. 1 – 4.

K.J. Larsen and R. Rietkerk, MULTIVARIATERESIDUES: A Mathematica package for computing multivariate residues, Comput. Phys. Commun. 222 (2018), 250 – 262.

P.-L. Ma, S.-F. Tian, L. Zou and T.-T. Zhang, The solitary waves, quasi-periodic waves and integrability of a generalized fifth-order Korteweg-de Vries equation, Waves Random Complex Media (2018), 1 – 17.

V.A. Melnikova, Algorithm for partition polinomials analytical derivation, Systems, Methods, Technologies 3(19) (2013), 112 – 116.

P. Natalini and P.E. Paolo, Remarks on Bell and higher order Bell polynomials and numbers, Cogent Mathematics 3 (2016), 1 – 15.

D. Pelinovsky, J. Springael, F. Lambert and I. Loris, On modified NLS, Kaup and NLBq equations: Differential transformations and bilinearization, J. Phys. A: Math. Gen. 30(24) (1997), 8705 – 8717.

F. Qi and M.-M. Zheng, Explicit expressions for a family of the Bell polynomials and applications, Appl. Math. Comput. 258 (2015), 597 – 607.

F. Qi, X.-T. Shi, F.-F. Liu and D.V. Kruchinin, Several formulas for special values of the bell polynomials of the second kind and applications, J. Appl. Anal. Comput. 7(3) (2017), 857 – 871.

F. Qi and B.-N. Guo, Explicit formulas for special values of the Bell polynomials of the second kind and for the Euler numbers and polynomials, Mediterr. J. Math. 14(3) (2017), 1 – 14.

J. Riordan, An Introduction to Combinatorial Analysis, Princeton University Press (2016).

S. Roman, The Umbral Calculus, Academic Press (1984).

J. Squire, J. Burby and H. Qin, VEST: Abstract vector calculus simplification in Mathematica, Comput. Phys. Commun. 185(1) (2014), 128 – 135.

R.P. Stanley, Enumerative Combinatorics, Volume 2, Cambridge University Press (1999).

J.-M. Tu, S.-F. Tian, M.-J. Xu, P.-L. Ma and T.-T. Zhang, Solitons, Backlund transformation, and Lax pair for the ((2+1))-dimensional Boiti-Leon-Pempinelli equation for the water waves, Computers and Mathematics with Applications 72(9) (2016), 2486 – 2504.

Y.-L. Wang, Y.-T. Gao, S.-L. Jia, G.-F. Deng and W.-Q. Hu, Solitons for a ((2+1))-dimensional variable coefficient Bogoyavlensky-Konopelchenko equation in a fluid, Modern Phys. Lett. B 31(25) (2017), 1 – 18.

W. Wang, Euler sums and Stirling sums, J. Number Theory 185 (2018), 160 – 193.

F.S. Wheeler, Bell polynomials, ACM SIGSAM Bulletin 21(3) (1987), 44 – 53.

J.-L. Zhao and F. Qi, Two explicit formulas for the generalized Motzkin numbers, J. Inequal. Appl. 2017(1) (2017), 1 – 8.

DOI: http://dx.doi.org/10.26713%2Fjims.v10i4.1046

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