Estimates on binomial sums of partition functions

Let $p(n)$ denote the partition function and define $p(n,k)=\sum_{j=0}^{k}\binom{n-j}{k-j}p(j)$ where $p(0)=1$. We prove that $p(n,k)$ is unimodal and satisfies $p(n,k) < \frac{2.825}{\sqrt{n}}\, 2^n$ for fixed $n\ge 1$ and all $1\le k\le n$. This result has an interesting application: the minimal dimension of a faithful module for a $k$-step nilpotent Lie algebra of dimension $n$ is bounded by $p(n,k)$ and hence by $\frac{3}{\sqrt{n}}\, 2^n$, independently of $k$. So far only the bound $n^{n-1}$ was known. We will also prove that $p(n,n-1)<\sqrt{n}\exp(\pi\sqrt{2n/3})$ for $n\ge 1$ and $p(n-1,n-1)<\exp (\pi\sqrt{2n/3} )$.