I am a masters student and I am interested in number theory. Due to lockdown I have a lot of time and I thought of reading a research paper in Number theory which is " Many Odd zeta values are irrational by " Stephane Fischler, Johannes Sprang and Wadim Zudilin.

I have a question on page 8 just after the (3.6)

My question is -> how authors deduced that $c_k,j \leq (2D)^{3Dn} ( n! / (k)^{ n+1} )^{s+1-3D}$ , for n large.

!inequalities in which I have question ]1

!defination of $c_{k,j}$]2

I am not able to understand how authors came to this conclusion. I think probably using $c_k, j$ 's definition one could get it. I can divide and multiply by $(2)^{3Dn} $ and use that s+1> 3D and that there are n+1 terms in denominator raised to exponent s+1 . These things indicate me that definition of $c_{j, n} $ would be used but I am not getting exact given inequality.

Edit -> Unfortunately there is one more question I am having. I am not able to derive the inequality which is just after the line Using (3.1) and Stirling Approximation .

Can someone please tell how it will be derived.

I have tried it many times.

Fischler, Stéphane; Sprang, Johannes; Zudilin, Wadim, Many odd zeta values are irrational, Compos. Math. 155, No. 5, 938-952 (2019). ZBL1430.11097.