Prove the (numericallyevident) proposition that \begin{equation} \Sigma_{i=0}^\infty f(i) = 1, \end{equation} where \begin{equation} f(i)= 2^{4 i6} q(i) \frac{\Gamma(3 i+\frac{5}{2}) \Gamma(5 i+2)}{3 \Gamma(i+1) \Gamma(2 i+3) \Gamma(5 i+\frac{13}{2})} \end{equation} and \begin{equation} q(i) = 185000 i^5 +779750 i^4 +1289125 i^3 +1042015 i^2 +410694 i+63000= \end{equation} \begin{equation} i (5 i (25 i (2 i (740 i+3119)+10313)+208403)+410694)+63000. \end{equation} This is the special case ($\alpha=0$) of eqs. (4)(6) in arXiv:1301.6617, and eqs. (2)(3) in arXiv:1303.1125.
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