I am reading an article "Well-poised hypergeometric service for diophantine problems of zeta values" by W. Zudilin. Consider the quantities defined here in pg. $617$ $$\tilde{F_n}:= \frac{1}{2}\sum_{t=1}^{\infty} \tilde{R_n}(t)\tag{1}$$
where $$\tilde{R_n}(t):=\frac{\mathrm{d^2}}{\mathrm{dt^2}} \left( (2t+n) \left(\frac{(t-1)...(t-n).(t+n+1)...(t+2n)}{(t(t+1)...(t+n))^2}\right)^3\right)$$ Then we have $$\tilde{F_n}=\tilde{u_n}\zeta(7)+\tilde{w_n}\zeta(5)-\tilde{v_n}$$ where $\tilde{u_n}, \tilde{w_n}, \tilde{v_n}$ are positive rationals satisfying a fourth order difference equation with characteristic polynomial: $$\lambda^4+9264 \lambda^3-12116166 \lambda^2-752300 \lambda-19683$$

Question: I need a tight upper bound or an asymptotic for $\tilde{F_n}$ as $n\to\infty$.

Any help would be highly appreciated. Thank you!

I am reading an article "Well-poised hypergeometric service for diophantine problems of zeta values" by W. Zudilin. Consider the quantities defined here in pg. $617$ $$\tilde{F_n}:= \frac{1}{2}\sum_{t=1}^{\infty} \tilde{R_n}(t)\tag{1}$$
where $$\tilde{R_n}(t):=\frac{\mathrm{d^2}}{\mathrm{dt^2}} \left( (2t+n) \left(\frac{(t-1)...(t-n).(t+n+1)...(t+2n)}{(t(t+1)...(t+n))^2}\right)^3\right)$$ Then we have $$\tilde{F_n}=\tilde{u_n}\zeta(7)+\tilde{w_n}\zeta(5)-\tilde{v_n}$$ where $\tilde{u_n}, \tilde{w_n}, \tilde{v_n}$ are positive rationals satisfying a fourth order difference equation with characteristic polynomial: $$\lambda^4+9264 \lambda^3-12116166 \lambda^2-752300 \lambda-19683$$

Question: I need a tight upper bound or an asymptotic for $\tilde{F_n}$ as $n\to\infty$.

I asked this as bountied question on MSE a week ago but unfortunately received no answer. Any help would be highly appreciated. Thank you!