Denominator Growth and a Liouville-Type Irrationality Criterion for Reciprocal Central Binomial Power Series

We study the series S(c)=∑k≥2Eₖ-cₖ, where Eₖ=k/42kk=(2k-1)2k-3k-2 (OEIS A033876) and (cₖ) is a strictly increasing sequence of positive integers; the case cₖ=k gives S=∑k≥1(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 Dₙ(c)=lcm(E₂c₂,,Eₙcₙ) and the reach of the Liouville truncation criterion. (1) If cₖ₊₁/cₖ>1 (e.g. cₖ= rᵏ, $r>1$), then $S(c)$ is irrational; for r≥2 this already follows from a theorem of Erdős (1975), so the new range is $1<r<2$. (2) For cₖ=kA with integer A≥1, truncation fails: log Dₙ(c)/nA+1 tends to an explicit constant CA>ln4, given in closed form via zeta values, with C₁=ln2+π²/8 and CA→ln4 as A→∞. (3) For cₖ=exp(k^β) with β>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 δₙ~ C√ln n/n with C=1/(2√ln2). For the original series S, we prove that its generating function is entire and not D-finite, and show numerically that S≠ p/q for all q<10⁵⁰⁴, that S satisfies no integer relation of norm below about 10¹⁰⁰ with 1,π,e,π²,e²,ln2,ζ(3),γ, and that its coefficients satisfy no polynomial recurrence of order at most 4 and degree at most 3. The irrationality of S remains open.