TITLE: A conjectural closed form for $\sum\limits_{n=0}^\infty\frac{n!\,(2n)!}{(3n+2)!}$
QUESTION [50 upvotes]: Let
$$S=\sum\limits_{n=0}^\infty\frac{n!\,(2n)!}{(3n+2)!},\tag1$$
its numeric value is approximately $S \approx 0.517977853388534047...$${}^{[more\ digits]}$
$S$ can be represented in terms of the generalized hypergeometric function:
$$S={_3F_2}\left(\frac12,1,1;\ \frac43,\frac53;\ \frac4{27}\right)\cdot\frac12.\tag2$$

Let $\sigma$ be the closed-form expression constructed from integers and elementary functions as follows:
$$\sigma=3\,\alpha\,\ln(2\,\alpha+1)-\sqrt{\beta\,}\arccos\gamma,\tag3$$
where
$$\alpha=\frac{\sqrt[3]{3\,}}{6\,\sqrt[3]{2\,}}\left(\sqrt[3]{9-\sqrt{69}}+\sqrt[3]{9+\sqrt{69}}\right),\tag4$$
$$\beta=\frac1{4\,\sqrt[3]{2\,}}\left(\sqrt[3]{25+3\,\sqrt{69}}+\sqrt[3]{25-3\,\sqrt{69}}\right)-\frac12,\tag5$$
$$\gamma=\frac1{6\,\sqrt[3]{2\,}}\left(\sqrt[3]{57\,\sqrt{69}-459}-\sqrt[3]{57\,\sqrt{69}+459}\right)+\frac12\tag6$$
are the unique real roots of the following cubic equations:
$$8\,\alpha^3-2\,\alpha-1=0,\tag7$$
$$64\,\beta^3+96\,\beta^2+36\,\beta-23=0,\tag8$$
$$8\,\gamma^3-12\,\gamma^2+16\,\gamma+11=0.\tag9$$
Equivalently,
$$\sigma = \frac{3\,p}{2}\,\ln\big(p+1\big)-\frac{1}{2}\sqrt{\frac{3-p}{p}}\arccos\Big( \frac{p-6}{6p+2}\Big)\tag{10}$$
where $p$ is the plastic constant or the real root of
$$p^3-p-1=0\tag{11}$$

It can be numerically checked that the following inequality holds:
$$\Big|S-\sigma\Big|<10^{-10^5},\tag{12}$$
I conjecture that the actual difference is the exact zero, and thus $S$ has an elementary closed form:
$$\sum\limits_{n=0}^\infty\frac{n!\,(2n)!}{(3n+2)!}\stackrel?=3\,\alpha\,\ln(2\,\alpha+1)-\sqrt{\beta\,}\arccos\gamma,\tag{13}$$
I am asking for you help in proving this conjecture.

REPLY [6 votes]: Alternatively, one can find a simple and elementary solution in logarithms as,
$$\sum\limits_{n=0}^\infty\frac{n!\,(2n)!}{(3n+2)!}=  -x_1\ln(1-x_1)-x_2\ln(1-x_2)-x_3\ln(1-x_3) = 0.5179778\dots$$
where the $x_n$ are the three roots of $x^3-x+1=0$
and which is the minpoly of the negated plastic constant. See also this post.