Let $\Gamma(x) := \int_0^\infty e^{-t}t^{x-1} \, \text d t$ denote the (Euler-)Gamma function.
For $0<\alpha<1/2$, I want to calculate the limit $$ \lim_{x\to\infty} \sum_{n=0}^\infty \sum_{k=0}^{\lfloor \alpha\,n\rfloor} \frac{\Gamma(x+\frac{n}{3})\,\Gamma(x+\frac{n+1}{3})\,\Gamma(x+\frac{n+2}{3})}{\Gamma(x)\,\Gamma(k+x+1)\,\Gamma(n-k+x+1)} $$ or at least find a small upper bound. (According to some computations with python using mpmath the result might well be $0$.)
EDIT: With $f(x_1,x_2,x_3,x_1+x_2+x_3;\,1/3,1/3,1/3):= \frac{(x_1+x_2+x_3)!}{x_1!\,x_2!\,x_3!\, 3^{x_1+x_2+x_3}}$ the probability mass function of the equiprobable trinomial distribution as in Wikipedia, the problem is equivalent to calculating $$ \lim_{\substack{x\to\infty\\x\in\mathbb N}} \sum_{\substack{k_1,k_2\in \mathbb N_0\\k_1 < \alpha(k_1+k_2)}} f(x-1,x+k_1,x+k_2;\, 1/3,1/3,1/3).$$
For any $x\in\mathbb N$, we have $$\sum_{k_1,k_2\in \mathbb N_0} f(x-1,x+k_1,x+k_2;\, 1/3,1/3,1/3)=1.$$