Resolving the Hermite coefficient conjectures in OEIS A381423, Comments (revision 8)

We prove Mike Sheppard's conjecture in OEIS A381423, Comments, revision 8, that the exponent of the largest signed coefficient of the physicists' Hermite polynomial has successive differences in {1,-3}. A coefficient-ratio calculation gives an explicit formula and proves uniqueness, square-root growth, and the two asserted difference-block lengths. We give exact quadratic indices for the longer blocks. The corresponding absolute-coefficient calculation proves the conjecture in A381524 and the three formulas for equal absolute maxima in A381737. It also gives the absolute mode explicitly, locates its two difference-block types, and determines the asymptotic ratios of successive maximal coefficients.