MathOverflow

In our lecture the professor wrote two different formulas for the Beta Function integral.
The first one is:$$\int_{0}^{\infty} \frac{x^{M-1}}{(1+x)^{M+N}} \, dx = \beta(M, N)$$
The second one is:$$\int_{0}^{\infty} \frac{y^{M-1}}{(ay + b)^{M+N}} \, dy = \frac{1}{a^M b^N} \beta(aM, bN)$$...

This is a continuation of Questions 508768 and 508774.
By Example 11 of this 2014 paper of Chu and Zhang, we have the identity$$\sum_{k=1}^\infty\frac{30k-11}{k^3(2k-1)\binom{2k}k}=4\zeta(3).\tag{0}$$For $x>1/2$, we define$$f(x):=\frac{(30x-11)\Gamma(x)^4}{4x(2x-1)\Gamma(2x)^2}.$$Clearly, $f(k)=(30k...

[EDITED mostly to report on the answer by Kevin Costello(and to improve the gp code at the end)]
I thank Nicolas Dupont for the following question(and for permission to disseminate it further):
I have a playlist with, say, $N$ pieces of music. While using the shuffle option (each such piece is played randomly at each step), I realized that, generally speaking, I have to hear quite a lot of times the same piece before the last one appears. It...

Let $K$ be a number field, $O_K$ its ring of integers, and $B = \operatorname{Spec} O_K$. Let $C/K$ be a smooth proper geometrically connected curve of genus $3$ with semistable reduction over $B$. Write$$\Delta_{\mathrm{st}}(C) := \frac{1}{[K:\mathbb...