Identity for central binomial coefficients On Wikipedia I came across the following equation for the central binomial coefficients:
$$
\binom{2n}{n}=\frac{4^n}{\sqrt{\pi n}}\left(1-\frac{c_n}{n}\right)
$$
for some $1/9<c_n<1/8$.
Does anyone know of a better reference for this fact than wikipedia or planet math? Also, does the equality continue to hold for positive real numbers $x$ instead of the integer $n$ if we replace the factorials involved in the definition of the binomial coefficient by Gamma functions?

Let
\begin{align*}
\mathbb{Z}&=\{0,\pm1,\pm2,\dotsc\}, & \mathbb{N}&=\{1,2,\dotsc\},\\
\mathbb{N}_0&=\{0,1,2,\dotsc\}, & \mathbb{N}_-&=\{-1,-2,\dotsc\}.
\end{align*}
The extended binomial coefficient $\binom{z}{w}$ for $z,w\in\mathbb{C}$ is defined by
\begin{equation}\label{Gen-Coeff-Binom}
\binom{z}{w}=
\begin{cases}
\dfrac{\Gamma(z+1)}{\Gamma(w+1)\Gamma(z-w+1)}, & z\not\in\mathbb{N}_-,\quad w,z-w\not\in\mathbb{N}_-;\\
0, & z\not\in\mathbb{N}_-,\quad w\in\mathbb{N}_- \text{ or } z-w\in\mathbb{N}_-;\\
\dfrac{\langle z\rangle_w}{w!},& z\in\mathbb{N}_-, \quad w\in\mathbb{N}_0;\\
\dfrac{\langle z\rangle_{z-w}}{(z-w)!}, & z,w\in\mathbb{N}_-, \quad z-w\in\mathbb{N}_0;\\
0, & z,w\in\mathbb{N}_-, \quad z-w\in\mathbb{N}_-;\\
\infty, & z\in\mathbb{N}_-, \quad w\not\in\mathbb{Z},
\end{cases}
\end{equation}
where
\begin{equation*}%\label{falling-Factorial}
\langle\alpha\rangle_n=\prod_{k=0}^{n-1}(\alpha-k)
=
\begin{cases}
\alpha(\alpha-1)\dotsm(\alpha-n+1), & n\in\mathbb{N}\\
1, & n=0
\end{cases}
\end{equation*}
is called the falling factorial.
Equation (10) on page 116 in the paper [1] below reads that the double inequality
\begin{equation}\label{Merkle-gamma-ineq}\tag{1}
6^x<\frac{\Gamma(2(1+x))}{\Gamma^2(1+x)}<(1+x)3^x
\end{equation}
holds for $x\in(0,1)$ and its reversed inequality is valid for $x>1$. We can reformulate the double inequality \eqref{Merkle-gamma-ineq} as follows: the double inequality
\begin{equation}\label{Gen-Central-Binom-Coeff-bounds-Eq}\tag{2}
\frac{6^x}{2x+1}>\binom{2x}{x}>\frac{(x+1)3^x}{2x+1}
\end{equation}
is valid for $x>1$ and its reversed version holds for $0<x<1$.
By the way, I have verified the following double inequalities:

*

*For $(0,+\infty)$, the double inequality
\begin{equation}\label{centr-binom-(2x+1)ineq}\tag{3}
\frac{2^{2x}}{2x+1}<\binom{2x}{x}<\frac{e^{2x}}{2x+1}
\end{equation}
is sharp in the sense that the bases $2$ and $e$ cannot be replaced by larger and smaller constants, respectively.
For $x\in\bigl(-\frac{1}{2},0\bigr)$, the right hand side inequality in \eqref{centr-binom-(2x+1)ineq} is still valid, but the left hand side inequality in \eqref{centr-binom-(2x+1)ineq} is reversed.

*For $x\in(0,+\infty)$, the double inequality
\begin{equation}\label{binom(2x+1)(x)-ineq}\tag{4}
e^x<\binom{2x+1}{x}<4^x
\end{equation}
is sharp in the sense that the bases $e$ and $4$ cannot be replaced by larger and smaller constants, respectively.
For $x\in\bigl(-\frac{3}{2},0\bigr)$, the left hand side inequality in \eqref{binom(2x+1)(x)-ineq} is still valid, but the right hand side inequality in \eqref{binom(2x+1)(x)-ineq} is reversed.

I am looking for a suitable outlet for publishing, among other closely related things, the double inequalities \eqref{centr-binom-(2x+1)ineq} and \eqref{binom(2x+1)(x)-ineq}.
References

*

*Milan Merkle, On log-convexity of a ratio of gamma functions, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. 8 (1997), 114--119; available online at https://www.jstor.org/stable/43666390.

*Feng Qi, Chao-Ping Chen, and Dongkyu Lim, Several identities containing central binomial coefficients and derived from series expansions of powers of the arcsine function, Results in Nonlinear Analysis 4 (2021), no. 1, 57--64; available online at https://doi.org/10.53006/rna.867047.

*Yue-Wu Li and Feng Qi, The sum of an alternating series involving central binomial numbers and its three proofs, Journal of the Korea Society of Mathematical Education Series B: The Pure and Applied Mathematics 28 (2021), no. 4, in press.

