I have found this limit in https://oeis.org/A019609 and I was wondering how to prove it (if it is actually correct):
$$\lim_{n\to\infty} \frac{4n}{a^2_n}=\pi e$$
where
$$a_1=0,a_2=1, a_{n+2}=a_{n+1}+\frac{a_n}{2n}.$$
By computer evaluation, it is correct for $2$ digits after decimal point at about $n\approx 24100$, so if it is correct, it converges really slow.
I've attempted to prove this by first considering generating function $f(x)=\sum_{n \geq 1}a_nx^n$ and then trying to get asymptotics of its coefficients. By using recurrence, we get $f(x)/x^2-1=f(x)/x+\sum \frac{a_n}{2n}x^n$, and after differentiation we get differential equation which solves to $$f(x)=\frac{e^{-x/2}x^2}{(1-x)^{3/2}}.$$ Now I think this is a step away from getting asymptotics of $a_n$, but I don't know how. Can anybody show how to finish this? Or maybe there is another way?
Also, I don't think it is useful, but here is at least closed form obtained from the $f(x)$ using binomial series and exponential function series:
$$
a_n=\sum_{i=0}^{n-2}\frac{(-1)^n}{2^i i!}\binom{-3/2}{n-i-2}.
$$
Closest to this question seems to be Mirror algorithm for computing $\pi$ and $e$ – does it hint on some connection between them?, where there are two sequences approaching $\pi$ and $e$ and solutions seem to use same approach using generating functions, so this seems to be on the right track.
Best Answer
$$a_n=\sum_{i=0}^{n-2}\frac{(-1)^n}{2^i i!}\binom{-3/2}{n-i-2}=(-1)^n \binom{-\frac{3}{2}}{n-2} \, _1F_1\left(2-n;\frac{3}{2}-n;-\frac{1}{2}\right)$$ $$\frac {4n}{a_n^2}=\frac{4 n}{\binom{-\frac{3}{2}}{n-2}^2 \,\,\Big[\, _1F_1\left(2-n;\frac{3}{2}-n;-\frac{1}{2}\right){}\Big]^2}$$
For large values of $n$ $$\frac{4 n}{\binom{-\frac{3}{2}}{n-2}^2}=\pi +\frac{5 \pi }{4 n}+\frac{49 \pi }{32 n^2}+O\left(\frac{1}{n^3}\right)$$
Now, computing the values of the Kummer's confluent hypergeometric function, the numerators are $$\{0,1,2,10,68,604,6584,85048,1269680,21505552\}$$ and they correspond to sequence $A165968$ in $OEIS$. For this, Vaclav Kotesovec proposed as an approximation $$2^{(n-\frac{1}{2})} e^{-(n+\frac{1}{2})} n^{(n-1)}$$
The denominators are the double factorials of odd numbers.
Using Vaclav Kotesovec' approximation for the numerator we end with $$\, _1F_1\left(2-n;\frac{3}{2}-n;-\frac{1}{2}\right)\sim \frac{2^{(n-\frac{1}{2})} e^{-(n+\frac{1}{2})} n^{(n-1)}}{(2n-3)!!}$$
Using Stirling like approximation for the denominator, $$\, _1F_1\left(2-n;\frac{3}{2}-n;-\frac{1}{2}\right)\sim \frac{1}{\sqrt{e}}-\frac{11}{24 \sqrt{e} n}-\frac{23}{1152 \sqrt{e} n^2}+O\left(\frac{1}{n^3}\right)$$ which finally makes $$\frac {4n}{a_n^2}=e\pi \left(1+\frac{13}{6 n}+\frac{241}{72 n^2} \right)+O\left(\frac{1}{n^3}\right)$$
Edit
I made a mistake trying to get the asymptotics of Kummer's function from its pieces.
Using what @Gary gave in comments, we end with $$\frac {4n}{a_n^2}=e\pi \left(1+\frac{3}{4 n}+\frac{13}{32 n^2}+\frac{1}{128 n^3}-\frac{669}{2048 n^4} \right)+O\left(\frac{1}{n^5}\right)$$ which is very different even for the expansion up to $O\left(\frac{1}{n^2}\right)$.