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Explore parametrized algebraic equations through an experimental mathematical approach in this 51-minute conference talk. Examine several instances of integer sequence pairs $(c(p,n), a(p,n))$ for integer $n\ge0$ and rational parameters $p\gt0$ that are interrelated through the exponential generating function relationship $\exp(\sum_{n=1}^{\infty}\,c(p,n).\frac{t^n}{n})=\sum_{n=0}^{\infty}\,a(p,n).t^n$. Learn about exact theoretical statements from Kontsevich and Reutenauer indicating that algebraicity of ordinary generating functions of $a(p,n)$ sequences implies algebraicity of generating functions of $c(p,n)$ sequences. Discover how to employ experimental-type approaches by constructing algebraic generating functions of $c(p,n)$ sequences that produce algebraic generating functions of $a(p,n)$ sequences, utilizing binomial coefficients parametrized by $p$ that result in generalized hypergeometric functions. Understand how to recover algebraicity of generating functions even without explicit knowledge of functional forms of $a(p,n)$, and derive novel logarithmic identities between various generalized hypergeometric functions when $a(p,n)$ sequences can be fully identified. Investigate the solution of corresponding Hausdorff moment problems using Meijer $G$-functions when conceiving $c(p,n)$ and $a(p,n)$ as moments, including analytical and graphical studies of weight function behavior in collaborative research with G. H. E. Duchamp, M. Kontsevich, and G. Koshevoy.
