Journal of Integer Sequences, Vol. 23 (2020), Article 20.1.3

Asymptotic Estimate for the Multinomial Coefficients

Jiyou Li
School of Mathematical Sciences
Shanghai Jiao Tong University
Shanghai, 200240


The multinomial coefficient $\binom{n,q}{k}$ is defined to be the coefficient of xk in $(1+x+x^2+\cdots
+x^{q-1})^n$. It is conjectured that for given n>2, $T(n, q):=\binom{n,q}{cn}-\binom{n,q-1}{cn}$ is unimodal and the maximum occurs at $q=\lfloor\log_{1+\frac 1{c}}{n}\rfloor$ or $q=\lfloor\log_{1+\frac 1{c}}{n}\rfloor+1$. As an attempt to prove this conjecture, we give an asymptotic estimate for $\binom{n,q}{cn}$ as n tends to infinity, where c is a positive integer.

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(Concerned with sequences A002426 A005725 A187925 A305161.)

Received January 29 2018; revised versions received February 4 2018; August 23 2018; June 8 2018; June 12 2018; September 5 2018; September 8 2018; May 30 2019; August 25 2019. Published in Journal of Integer Sequences, December 28 2019.

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