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I have noticed that if we take the Laurent expansion of the Riemann zeta function about $s=1$ $$ \zeta(s) = \frac{1}{s-1} + \sum_{n=0}^\infty \frac{(-1)^n}{n!}\gamma_n (s-1)^n $$ which defines $\gamma_n$, the Stieltjes constants, where $\gamma_0$ is the Euler-Mascheroni constant. Next perform a series reversion on this to give a series $$ \chi(s) = 1+\frac{1}{s}+\sum_{n=2}^\infty \frac{\kappa(n)}{s^n} $$ which has expansion \begin{equation} \chi(s) = 1 + \frac{1}{s} + \frac{\gamma_0}{s^2} + \frac{\gamma_0^2-\gamma_1}{s^3} + \frac{2\gamma_0^3-6\gamma_0\gamma_1+\gamma_2}{2s^4} + \cdots \label{expansion} \end{equation}

The coefficients $\kappa(n)$ seem to decrease quite steadily, even up $n$ being a few hundred, where the $\gamma_n$ get large.

\begin{array}{ |c|c|c| } \hline n & \kappa(n) \\ \hline 0 & 1.000000000\\ 1 & 1.000000000\\ 2 & 0.5772156649\\ 3 & 0.4059937693\\ 4 & 0.3135616752\\ 5 & 0.2556464523\\ 6 & 0.2159181431\\ 7 & 0.1869526867\\ 8 & 0.1648872027\\ 9 & 0.1475121704\\ 10 & 0.1334717457\\ 11 & 0.1218874671\\ 12 & 0.1121649723\\ 13 & 0.1038876396\\ 14 & 0.09675470803\\ 15 & 0.09054358346\\ \hline \end{array}

Letting $$ R_n=\sum_{k=1}^n i_k $$ and $$ P_n=\sum_{k=1}^n ki_k $$ and $\{i\}_n=\{i_1,i_2,\cdots|P=n-1\}$, I have observed the expression for $\kappa(n)$ from series reversion to be $$ \kappa(n)=\sum_{\{i\}} (-1)^n\left[\prod_{j=1}^{R-1}j-n\right]\left[\prod_{k=1}^n \frac{1}{i_k!}\left(\frac{\gamma_k}{k!}\right)^{i_{k+1}}\right]\gamma_0^{i_1} $$ where we define $\kappa(0)=1$. Two examples $$ \kappa(3) = \gamma_0^2 - \gamma_1 = -\sum_{i_1+2i_2+3i_3=2} \left[\prod_{j=1}^{i_1+i_2+i_3-1} (j-n)\right]\frac{\left(\frac{\gamma_1}{1!}\right)^{i_2}\left(\frac{\gamma_2}{2!}\right)^{i_3}}{i_1!i_2!}\gamma_0^{i_1} $$ $$ \kappa(4) = \gamma_0^3 - 3 \gamma_0\gamma_1 + \frac{\gamma_2}{2} = \sum_{i_1+2i_2+3i_3+4i_4=3} \left[\prod_{j=1}^{i_1+i_2+i_3+i_4-1} (j-n)\right]\frac{\left(\frac{\gamma_1}{1!}\right)^{i_2}\left(\frac{\gamma_2}{2!}\right)^{i_3}\left(\frac{\gamma_3}{3!}\right)^{i_4}}{i_1!i_2!i_3!}\gamma_0^{i_1} $$

Can we prove or disprove that $$ \kappa(n+1) < \kappa(n), \;\;\; n\in\mathbb{N}^{>0}? $$ Is this perhaps a more well behaved way to look at the Stieltjes constants? Any explanations to why this happens and whether it is noteworthy?

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