And in fact the coefficients you obtained with the code are not correct because of integer overflow. (I got these coefficients from Mathematica.)
julia> compute_hermite_polynomial(48) Poly(8629227277223198720 - 8374539685103403008x^2 + 2715657340094251008x^4 + 6791467061357838336x^6 + 8259543481672794112x^8 + 9055289415143784448x^10 + 376270965132230656x^12 + 1067414392280186880x^14 + 8920937959042056192x^16 - 4036022274615148544x^18 + 6462554277163302912x^20 - 783339912383430656x^22 + 7024739932564357120x^24 + 6859947603694452736x^26 - 5181745414033899520x^28 - 5062547358466703360x^30 - 634480881389535232x^32 + 5903955228692054016x^34 - 3820547819823955968x^36 - 5156551204595040256x^38 + 2667573538658975744x^40 - 1674494636451692544x^42 + 4183562578850480128x^44 - 158751886864809984x^46 + 281474976710656x^48)
