The recursion relationship: zPn(z) = Pn+1(z) + 𝛽nPn−1(z), n = 0, 1, 2 … is satisfied by all monic orthogonal polynomials in regard to an arbitrary Freud-type weight function. In current paper, one focuses on the weight function 𝜔(z) = |z| 𝛼e−z6+tz2 , z ∈ R, t ∈ R, 𝛼> −1 to analyze its relative 𝛽n and Pn(z). Through above equation and orthogonality, we find that 𝛽n(t) satisfy the first discrete Painlevé equation I Hierarchy and a high-order differential-difference equation, respectively. Then, we find that the asymptotic value of 𝛽n is settled by Coulomb fluid. Additionally, we talk about Pn(z) with 𝛼 = 0, including approximation for Pn(0) and P′ n(0), and bounds for Pn(z) as n → ∞ are settled.