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It is not clear to me what you mean by "giving an expression for the n-th coefficient". Is this what you are looking for?

sage: R.<z> = FractionField(QQ[['z']])
sage: R
Laurent Series Ring in z over Rational Field
sage: (z^-2 + z^3 + 5*z^8 + O(z^10)).integral()
-z^-1 + 1/4*z^4 + 5/9*z^9 + O(z^11)