Series for Stieltjes constants from $\gamma= \sum_{n=1}^\infty \left(\frac{2}{n}-\sum_{j=n(n-1)+1}^{n(n+1)} \frac{1}{j}\right)$

Euler’s constant has the following representations (Euler-Mascheroni constant expression, further simplification, https://math.stackexchange.com/a/129808/134791, Question on Macys formula for Euler-Mascheroni Constant $\gamma$)

$$
\gamma= \lim_{n \to \infty} {\left(2H_n-H_{n^2} \right)}=\sum_{n=1}^\infty \left(\frac{2}{n}-\sum_{j=(n-1)^2+1}^{n^2} \frac{1}{j}\right)
$$

$$
\gamma= \lim_{n \to \infty} {\left(2H_n-H_{n(n+1)} \right)}=\sum_{n=1}^\infty \left(\frac{2}{n}-\sum_{j=n(n-1)+1}^{n(n+1)} \frac{1}{j}\right)
$$

$$
\gamma= \lim_{n \to \infty} {\left(2H_n-\frac{1}{6}H_{n^2+n-1}-\frac{5}{6}H_{n^2+n}\right)}$$$$=\frac{7}{12}+\sum_{n=1}^{\infty}\left(\frac{2}{n+1}-\frac{1}{3n(n+1)(n+2)}-\sum_{k=n(n+1)+1}^{(n+1)(n+2)}\frac{1}{k}\right)
$$

Hardy (1912) and Kluyver (1927) derived formulas for $\gamma_1$ and $\gamma_n ,n>1$ from Vacca’s formula for $\gamma$ (1910) (http://mathworld.wolfram.com/StieltjesConstants.html)

Q: How can formulas for Stieltjes constants be derived from the formulas above?

(See also Re-Expressing the Digamma)

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