Exact Closed-Form Hydrogenic Radial Expectation Values $⟨r^k⟩$ for $k = -2$ to $-30$.

This dataset provides exact closed-form symbolic expressions for the non-relativistic hydrogenic radial expectation values ⟨rᵏ⟩ across the integer range k = −2 to −30. The expressions are valid for arbitrary principal quantum number n, orbital angular momentum ℓ, and nuclear charge Z, and are presented as rational functions with explicit polynomial numerators and factorized denominators. All results are derived analytically without numerical integration or quadrature approximations. Lower-order moments (k = −2, −3, −4) reproduce standard textbook results, while k = −5 to −8 are benchmarked against established analytical references. For higher negative powers (k ≤ −9), independent validation is performed using the Kramers–Pasternack recurrence relation implemented in Mathematica. A dual-seed recursion strategy (k = −2, −3 extended to −18; k = −16, −17 extended to −30) confirms term-by-term exact agreement after normalization alignment. The dataset is provided in LaTeX and Mathematica-compatible formats for direct use in precision atomic physics, perturbative calculations, and pedagogical applications. It serves as a supplementary reference companion to a forthcoming unified analytical framework for hydrogenic expectation values.