Estimates on binomial sums of partition functions

Dietrich Burde

Let $p(n)$ denote the partition function and define $p(n,k)={\sum }_{j=0}^k\left(\genfrac{}{}{0ex}{}{n-j}{k-j}\right)p(j)$ where $p(0)=1$. We prove that $p(n,k)$ is unimodal and satisfies $p(n,k) < \frac{2.8…