I guess that the following was solved sometime in the 18th century, but could not find a reference to it. I am interested in approximations to the following integer partition problem:
Denote $R(N,L)$ as the number of ways to write an integer L as a sum (might be empty) of distinct positive integers in the range [1..N] (where order is unimportant). Formally
$R(N,L) = \sum_{ \{\sigma_n\}_{n=1}^{N} } I\left( \sum_{n=1}^N \sigma_n n = L \right) $
I know that $R(N,L)$ satisfies
$R(N,L) = R(N-1,L) + R(N-1,L-N)$,
and $R(N,0)=1$ for all $N$.
In addition, I know that it has several entries in Sloane's OEIS, for instance A053632 (https://oeis.org/A053632), as the coefficients of the product $\prod_{n=1}^N (1+x^n)$.
What I am looking for is some simple expression which approximates it for large enough $N$ (but finite!). From playing with it in Matlab I found that
$R(N,L) \approx A\times 2^{\sqrt{\sin \left( \frac{2\pi L}{N(N+1)} \right)} }$
But I am not sure what $A$ is, whether this is really the asymptotic for large $N$, and what is the finite $N$ corrections.
If anyway has a clue, I will be grateful.
Many thanks,
Amir Bar