Type X Ramanujan-type series identity for 1/π with T_k(19,−20) and T_{2k}(9,−5)

Prove that the infinite series sum_{k=0}^∞ [(16k+3)/(-202^2)^k] * binom(2k,k) * T_k(19,-20) * T_{2k}(9,-5) equals 43√101/(75π), where T_n(b,c) denotes the coefficient of x^n in (x^2 + b x + c)^n.

Background

The paper collects over 100 conjectural series identities involving binomial-type coefficients, many of which are Ramanujan-type series for 1/π. The example highlighted in the abstract showcases a new identity of the newly introduced Type X, where the summand mixes generalized central trinomial coefficients T_k(b,c) and T_{2k}(b_,c_).

The generalized central trinomial coefficient T_n(b,c) is defined as the coefficient of x^n in (x^2 + b x + c)^n and generalizes the central binomial coefficient via T_n(2,1)=binom{2n}{n}. The conjectured identity gives a rapidly convergent series evaluation to a rational multiple of 1/π times an algebraic factor, typical of Ramanujan-type formulas.