Number of partitions of $2n$ with no element greater than $n$ The number of partitions of $2n$ into partitions with no element greater than $n$ (copied and slightly adapted from http://mathworld.wolfram.com/PartitionFunctionq.html), so I'm looking for a nice formula of $q(2n,n)$.
By asking http://www.wolframalpha.com/input/?i=integer+partitions+of+12 and counting the ones of interest I get results from 2 to 12, that look the following:
$$
1,3,7,15,30,58
$$
which matches the data from http://oeis.org/A026820/table when you start from the $1$ in the 2.row and go straight down. My question is now, if there is closed formula or at least an asymptotic for $q(2n,n)$?
 A: 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 
$$q(2n,n)\sim p(2n) \sim \frac {1} {8n\sqrt{3}} e^{\pi \sqrt {\frac{4n}{3}}}\;.$$
A: 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.
A: Using the following result for restricted partition generating functions, what I need can be calculated like 
$$
\left(\frac{d^{2n}}{dx^{2n}}\prod_{k=1}^n (1-x^k)^{-1} \right) \Biggr|_{x=0}
=\left(\frac{d^{2n}}{dx^{2n}}\sum_{k=0}^\infty h_{nk}x^k\right) \Biggr|_{x=0},
$$
where $h_{nk}$ is the number of ways, that $k$ can be written with $n$ as largest value.
