bibkey: lin2018catalanpowers authors: Gwo Dong Lin year: 2018 title: “On Powers of the Catalan Number Sequence” doi: 10.1016/j.disc.2018.05.009 url: https://arxiv.org/abs/1711.01536v3 claim: “Equation (3), Theorem 1, and Lemma 2 provide the Catalan moment law and the product-density route for powers of Catalan moments.” strata_touched:

• D5/S3/Constants/Moments/CatalanSquareHankelGrowth
• D5/S3/Constants/Moments/CatalanSquareHankelLimits license: citation-only triage: anchor

Lin’s Catalan moment and product laws

Verified locator

The supplied primary-source audit read https://arxiv.org/abs/1711.01536v3 (DOI 10.1016/j.disc.2018.05.009) and identifies page 3, Equation (3), page 4, Theorem 1, and page 7, Lemma 2 as the relevant passages. Equation (3) gives the classical Catalan probability density on (0,4). Theorem 1 treats positive powers of the Catalan moment sequence, and Lemma 2 supplies the density formula for a product of independent positive random variables.

Consequently, if independent random variables X and Y have the Catalan law, then Z=X*Y is supported on (0,16), has moments E[Z^m]=C_m^2, and has a density positive throughout that open interval. The formal proof uses the equivalent scaled beta law directly and evaluates its moments inside Lean. It does not import Lin’s analytic derivation as a formal premise.

Scope

The equation and theorem locators above are inherited from the supplied source audit; this metadata pass did not independently reread the paper. The note attributes the classical moment/product route and does not claim that Lin states either OEIS Hankel limit or the repository’s elementary Chebyshev-Gram proof.