DENOMINATOR GROWTH AND A LIOUVILLE-TYPE IRRATIONALITY CRITERION FOR RECIPROCAL CENTRAL BINOMIAL POWER SERIES

We study the series $S(c)=\sum_{k\ge2}E_k^{-c_k}$, where $E_k=\frac{k}{4}\binom{2k}{k}=(2k-1)\binom{2k-3}{k-2}$ (OEIS A033876) and $(c_k)$ is a strictly increasing sequence of positive integers; the case $c_k=k$ gives $S=\sum_{k\ge1}(2B(k,k))^k$, built from central values of the Euler Beta function. Using Kummer's theorem and the prime number theorem, we determine the growth of the denominators $D_n(c)=\mathrm{lcm}(E_2^{c_2},\dots,E_n^{c_n})$ and the reach of the Liouville truncation criterion. (1) If $\liminf c_{k+1}/c_k>1$ (e.g. $c_k=\lceil r^k\rceil$, $r>1$), then $S(c)$ is irrational; for $r\ge2$ this already follows from a theorem of Erdős (1975), so the new range is $1<r<2$. (2) For $c_k=k^A$ with integer $A\ge1$, truncation fails: $\log D_n(c)/n^{A+1}$ tends to an explicit constant $C_A>\ln4$, given in closed form via zeta values, with $C_1=\ln2+\pi^2/8$ and $C_A\to\ln4$ as $A\to\infty$. (3) For $c_k=\lceil\exp(k^\beta)\rceil$ with $\beta>1/2$, truncation succeeds and $S(c)$ is irrational. We conjecture, on heuristic grounds supported by exact-valuation computations, that the transition occurs at relative increments $\delta_n\sim C\sqrt{\ln n/n}$ with $C=1/(2\sqrt{\ln2})$. For the original series $S$, we prove that its generating function is entire and not D-finite, and show numerically that $S\ne p/q$ for all $q<10^{504}$, that $S$ satisfies no integer relation of norm below about $10^{100}$ with $1,\pi,e,\pi^2,e^2,\ln2,\zeta(3),\gamma$, and that its coefficients satisfy no polynomial recurrence of order at most 4 and degree at most 3. The irrationality of $S$ remains open.