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We encounter with following integral in one of our research paper, we calculate this integral numerically but it seems that this integral may be solved in Confocal Ellipsoidal Coordinates analytically, any suggestion for solving analytically this integral is appreciated. ‎$Q(\xi)=\int_0^{2\pi} \int_0^{\pi} \int_0^{2a_0} \left(e^{-\frac{\rho}{a_0}}+e^{-\frac{\sqrt{\rho^2-2\xi \rho \sin \theta \sin \phi+\xi^2}}{a_0}}\right)^2{\rho}^2 \sin \phi d \rho d \phi d \theta‎$

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Calculations with Maple suggest that $$ Q(\xi) = \pi a_0^3\sum_{k=0}^\infty \frac{c_k+d_ke^{-4}}{(2k+1)!}(\xi/a_0)^{2k} $$ where the coefficients $c_k$ and $d_k$ are integers. The first few values can be tabulated as follows: $$ \begin{array}{|c|c|c|c|c|c|c|}\hline k & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ \hline c_k & 4 & -2 & -22 & -138 & -782 & -4114& -20502 & -98330 \\ \hline d_k & -52 & -38 & -34 & 162 & 1846 & 12362 & 69726 & 360562 \\ \hline \end{array} $$ OEIS does not recognize these sequences. It looks like $c_k/c_{k-1}$ and $d_k/d_{k-1}$ might both converge to the same limit of approximately 4.5, but one would have to calculate more terms to get convincing evidence of that.

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