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On the Coefficients of the Distinct Monomials in the Expansion of
***x*_{1}(*x*_{1}+*x*_{2}) ··· (*x*_{1}+*x*_{2}+ ··· +*x*_{n})

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Sela Fried

Department of Computer Science

Ben-Gurion University of the Negev

David Ben Gurion Blvd 1

Be'er Sheva

Israel

**Abstract:**

We initiate the study of the coefficients of the distinct
monomials in the expansion of the multivariate polynomials
*x*_{1}(*x*_{1}+*x*_{2}) ··· (*x*_{1}+*x*_{2}+ ··· +*x*_{n}),
*n* ∈ **N**, the number of which was
shown by Shallit to be counted by the Catalan numbers
*C*_{n}, *n* ∈ **N**. In
particular, we obtain an exact formula for the coefficients and reduce
the complexity of the search for their maximum from the order of
*C*_{n}
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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