\[ \large \displaystyle\sum _{ n=1 }^{ \infty }{ \dfrac { { H }_{ n }-\ln { n } -\gamma }{ n } } =-\dfrac { { \gamma }^{ t } }{ a } -h{ \gamma }_{ i }+\dfrac { { \pi }^{ n } }{ { i }_{ 1 } } \]

The equation above holds true for positive integers \(t,a,h,i, n\) and \(i_1\). Find \(t+a+h+i+n+i_1\).

**Notations**:

- \(H_n\) denotes the \(n^\text{th} \) harmonic number.
- \( \gamma\) denotes the Euler-Mascheroni constant, \(\gamma \approx 0.5772 \).
- \(\gamma_n\) denotes the \(n^\text{th} \) Stieltjes constant.

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