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We have the following recursion $$ C_{g,n+1}= \left(n+1 C_{g-1,n+2}+\sum_{g_1 +g_2 = g\atop n_1+n_2 =n +1}^{\text{stable}}C_{g_1 , n_1 +1}C_{g_2 , n_2 +1} \right)\frac{(D_{g,n}+1)^{(D_{g,n}+1)}}{D_{g,n+1}^{D_{g,n+1}}}+2C_{g,n}\frac{(2D_{g,n}+1)^{(2D_{g,n}+1}}{27(D_{g,n+1}-2)^{(2D_{g,n+1}}-2)} $$ with the bases cases $C_{0,3}=1 , C_{1,1}=1, C_{g,0}=0$ and stable imply $g_i ,n_i\neq (0,1), (0,2)$.

In the following paper https://arxiv.org/pdf/1905.11270.pdf

We have the theorem 2.2 claiming to abound for the number $C_{g,n}$ $$C_{g,n}\leq 9(5g-5+n)! \exp(4g-4+3n)3^{5g-5+n}14^{-g} $$

It was proven in the paper by induction. My first question is how to get such bound in the first place. So given a recursion and bases cases is there is general theory giving such a bound

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