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
Israel
Abstract:
We initiate the study of the coefficients of the distinct
monomials in the expansion of the multivariate polynomials
x1(x1+x2) ··· (x1+x2+ ··· +xn),
n ∈ N, the number of which was
shown by Shallit to be counted by the Catalan numbers
Cn, 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
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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