Quasimodular forms arising from Jacobi's theta function and special symmetric polynomials

Ramanujan derived a sequence of even weight $2n$ quasimodular forms U₂ₙ(q) from derivatives of Jacobi's weight $3/2$ theta function. Using the generating function for this sequence, one can construct sequences of quasimodular forms of all nonnegative integer weights with minimal input: a weight 1 modular form and a power series $F(X)$. Using the weight 1 form θ(q)² and F(X)=exp(X/2), we obtain a sequence ₙ(q)\ of weight n quasimodular forms on Γ₀(4) whose symmetric function avatars Yₙ(xᵏ) are the symmetric polynomials Tₙ(xᵏ) that arise naturally in the study of syzygies of numerical semigroups. With this information, we settle two conjectures about the Tₙ(xᵏ). Finally, we note that these polynomials are systematically given in terms of the Borel-Hirzebruch A-genus for spin manifolds, where one identifies power sum symmetric functions pᵢ with Pontryagin classes.