We use computer algebra to-expand the Pekeris secular determinant for two-electron atoms symbolically, to produce an explicit polynomial in the energy parameter epsilon, with coefficients that are polynomials in the nuclear charge Z. Repeated differentiation of the polynomial, followed by a simple transformation, gives a series for epsilon in decreasing powers of Z. The leading term is linear, consistent with well-known behavior that corresponds to the approximate quadratic dependence of ionization potential on atomic number (Moseley's law). Evaluating the 12-term series for individual Z gives the roots to a precision of 10 or more digits for Z greater than or equal to 2. This suggests the use of similar tactics to construct formulas for roots vs atomic, molecular, and variational parameters in other eigenvalue problems, in accordance with the general objectives of gradient theory. Matrix elements can be represented by symbols in the secular determinants, enabling the use of analytical expressions for the molecular integrals in the differentiation of the explicit polynomials. The mathematical and computational techniques include modular arithmetic to handle matrix and polynomial operations, and unrestricted precision arithmetic to overcome severe digital erosion. These are likely to find many further applications in computational chemistry.