Wolfram wishlist: series of $\Gamma(z)$ in $z=-m$. Cycle index of symmetric groups.

Question

I found out that Wolfram has a wish list of formulas it is researching.

The first point is "Series for the gamma function"
We are searching for general formulas for the series expansion of the gamma function $\Gamma(z)$ near its regular points and poles $z=-n / n==0,-1,-2,...$. These formulas should include the psi­functions and their derivatives.

I calculated it: for $-m=0,-1,-2,...$ $$\Gamma(z)=\frac{(-1)^m}{m!(z+m)}\sum_{n=0}^{\infty}\frac{B_n(a_1,...,a_n)}{n!}(z+m)^{n}$$ where $$a_n=(n-1)!\left(H_m^{(n)}+(-1)^n\zeta(n)\right)\qquad\text{where:}\quad\zeta(1)=\lim_{h\to 0}\frac{\zeta(1+h)+\zeta(1-h)}{2}=\gamma$$ equivalently: $$a_n=(n-1)!H_m^{(n)}+\psi^{(n-1)}(1)$$ Where:

1. $B_n(a_1,...,a_n)$ is the $n$-th complete Bell polynomial
2. $H_m^{(n)}$ is the generalized harmonic number
3. $\zeta(z)$ is the Riemann zeta function
4. $\psi^{(\nu)}(z)$ is the polygamma function

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Question (I don't think it's possible) Does anyone have any ideas for writing this formula $$B_n\left(H_m+\psi(1),...,(n-1)!H_m^{(n)}+\psi^{(n-1)}(1)\right)$$ without using Bell polynomials (and without writing a very heavy formula)?

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PS: (this is a little off topic) I tried to send an email to the link they had put to provide contributions "[email protected]"
But I received an email reply with written The recipient server did not accept our requests to connect.
Does anyone happen to know some other way to contact them? So I send it to them.

Update

Considering $$b_n=H_m^{(n)}+(-1)^n\zeta(n)$$

It is possible to write the series as

$$\Gamma(z)=\frac{(-1)^m}{m!(z+m)}\sum_{n=0}^{\infty}Z(S_n)\cdot (z+m)^{n}$$

Where:

$\displaystyle Z(S_n):=\frac{B_n(0!\cdot b_0, 1!\cdot b_1, ..., (n-1)! \cdot b_{n-1})}{n!}$ is the cycle index of the symmetric group $S_{n}$