Journal of Integer Sequences, Vol. 25 (2022), Article 22.4.2

On the Coefficients of the Distinct Monomials in the Expansion of x1(x1+x2) ··· (x1+x2+ ··· +xn)

Sela Fried
Department of Computer Science
Ben-Gurion University of the Negev
David Ben Gurion Blvd 1
Be'er Sheva


We initiate the study of the coefficients of the distinct monomials in the expansion of the multivariate polynomials x1(x1+x2) ··· (x1+x2+ ··· +xn), nN, the number of which was shown by Shallit to be counted by the Catalan numbers Cn, nN. In particular, we obtain an exact formula for the coefficients and reduce the complexity of the search for their maximum from the order of Cn to the order of the number of partitions of n with distinct parts.

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(Concerned with sequences A000009 A000108 A000142 A001563 A052571 A062119 A144186 A144187 A299504 A347917 A349404.)

Received November 20 2021; revised versions received November 21 2021; March 28 2022. Published in Journal of Integer Sequences, March 30 2022.

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