Let $$f(k) := \frac{2k-1}{k}\bigl(1-\sum\limits_{i\lt k}\frac{i\ f(i)}{k+i-1}\bigr)$$ for $k\in\mathbb{N}^{+}$.
So $f(1) = 1$, $f(2) = 3/4$, $f(3) = 35/72$, etc.
Numerically it appears that $f(n)$ is approximately $1/(n\ ln(2))$. Can anyone give me any hints as to how one might try to prove that, or even a bound like $f(n) < 2/n$?