Alternating signs of the even-index Bernoulli-Catalan numbers τ_{2n}^*

Prove that the even-index special parameters τ_{2n}^*, defined by the s_i=0 specialization in the triangular transformation relating the series parameters τ_k and the integration constants s_k for the Bernoulli-Catalan polynomial family Q_n(z) generated by Q_1(z)=z−1/2 and Q_n(z)=K(∂_z) Σ_{k=1}^{n−1} Q_k(z) Q_{n−k}(z) with K(t)=t^{-1} tanh(t/2), alternate in sign with n (i.e., sign(τ_{2n}^*) = (−1)^n).

Background

Section 5 defines the Bernoulli-Catalan polynomials Q_n(z) through a Catalan-type nonlinear recurrence involving the differential operator K(∂_z)=tanh(∂_z/2)/∂_z, obtained in the s_i=0 specialization of the outer expansion formulation. The associated special parameters τ_k^* (the Bernoulli-Catalan numbers) appear via the triangular change of variables between τ and s and are explicitly listed for low indices.

The authors observe empirical alternation for τ_{2n}^* analogous to the Bernoulli-number sign pattern and state their conjecture that this alternation holds in general. Establishing this would clarify the sign structure of the τ-parameters underlying Q_n(z) and further connect them to classical Bernoulli phenomena.