Maximum of twisted binomial coefficients

For any integer $n$, define $$\mu(n)=\text{arg max}_{1\leq k\leq n}\binom{\frac{n+k}{2}}{k},$$ where the binomial coefficients are set to $0$ if $n+k$ is odd.

Question: Is the sequence $\left(\mu(n)\right)_{n\geq 1}$ known to some of you? In particular, I am interested in the asymptotic behaviour of $\mu(n)$ when $n$ goes to infinity.

More info: The first terms are as follow: $$1, 2, 1, 2, 3, 2, 3, 4, 5, 4, 5, 6, 5, 6, 7,\ldots$$ This sequence (or any subset) does not seem to be part of the OEIS. Moreover, it does not seem to have any periodic behaviour, that is, $\forall n\geq 1,\;\mu(n+p)=\mu(n)+q$. The two graphs below are plots of $\mu(n)$ for $n$ ranging from $1$ to $100$ and $1000$. The asymptotic does not seem linear (the red line is $0.45n$). I also used Stirling's formula to get an equivalent of $\binom{\frac{n+k}{2}}{k}$ when $n$ is large and $k=\alpha n$ with $\alpha \in (0,1)$ a constant, but it seems that there is no simple $\alpha^{\star}$ that maximizes this quantity.  • Mustn't $\mu(2)$ be even? I think $\mu(2)=2$, not $1$. – Théophile May 3 '16 at 12:56
• You are correct, I edited my post. – DGarreau May 3 '16 at 13:13