Number of partitions of $2n$ with no element greater than $n$

For large $n$, these are almost all the partitions there are. There can be at most one part $m$ larger than $n$, and the remaining parts form a partition of $2n-m$, so we have

$$q(2n,n)=p(2n)-\sum_{k=0}^{n-1}\;p(k)\;.$$

For large $n$, the terms in the sum are exponentially smaller than $p(2n)$, so asymptotically

$$q(2n,n)\sim p(2n) \sim \frac {1} {8n\sqrt{3}} e^{\pi \sqrt {\frac{4n}{3}}}\;.$$

The first $36$ values are:

1, 3, 7, 15, 30, 58, 105, 186, 318, 530, 863, 1380, 2164, 3345, 5096, 7665, 11395, 16765, 24418, 35251, 50460, 71669, 101050, 141510, 196888, 272293, 374423, 512081, 696760, 943442, 1271527, 1706159, 2279700, 3033772, 4021695, 5311627

Here’s a (lin-log) graph, also showing the curve for Joriki’s asymptotic expression $\frac{1}{8 n \sqrt{3}} e^{\pi \sqrt{\frac{4 n}{3}}}$.

$\hspace{1in}$

The list was generated using

Length@IntegerPartitions[2n, All, Range[n]]

in Mathematica.