Efficient method to approximate Euler sums to high precision

Question

I'm trying to approximate four weight $9$ Euler sums with high precision (say over $100$ digits) with Mathematica: $$\sum _{n=1}^{\infty } \frac{H_n^7}{n^2}, \sum _{n=1}^{\infty } \frac{H_n^5 H_n^{(2)}}{n^2}, \sum _{n=1}^{\infty } \frac{H_n^3 (H_n^{(2)})^2}{n^2}, \sum _{n=1}^{\infty } \frac{H_n^4 H_n^{(3)}}{n^2}$$ Commands I used are 'NSum' and 'WorkingPrecision'. However, the system failed to give an answer as I set the precision to be more than 50 digits.

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Can somebody help me approximate these sums to $100$ digits? Any kind of help will be appreciated.

Answer 1

A simple approach to make the software working is to replace \begin{align}H_n&=\gamma+\psi(n+1),\\H_n^{(k+1)}&=\zeta(k+1)+(-1)^k\psi^{(k)}(n+1)/k!,\end{align} which also "enables" E-M or A-P. The PARI/GP's sumnum "eats" it readily, producing

9043.5457472804393012140336705009063803732500237462521720823751494909142839177775416429118400246572080585...
355.94366198574004821936879947387876635374807856929634213045666004180991906871389314621941008708634541009...
28.236409513469581058820149577025324285151921286209876113436627856083937411911651120050250992517528455771...
54.648104639270712275307375212263275727767707498351542016967915613175186045675893730980305009885779661998...