On a Conjecture of Andrica and Tomescu

TL;DR

This paper proves a conjecture by Andrica and Tomescu, showing that a specific sequence's terms grow asymptotically like a scaled exponential function times a polynomial factor as n becomes large.

Contribution

The paper establishes the asymptotic behavior of the sequence S(n), confirming a conjecture and providing a precise growth rate for the sequence.

Findings

S(n) is asymptotic to /.14 2^n n^{-3/2} as n   infinity

The sequence's growth rate is characterized precisely for large n

Confirms the conjecture of Andrica and Tomescu

Abstract

Let S(n) be the integer sequence which is the coefficient of x^{n(n+1)/4} in the expansion of (1+x)(1+x^2), ..., (1+x^n) for positive integers n congruent to 0 or 3 mod 4. We prove a conjecture of Andrica and Tomescu that S(n) is asymptotic to \sqrt{6/\pi} 2^n n^{-3/2} as n approaches infinity.