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The higher rank Racah algebra $R(n)$ introduced recently is recalled. Aquotient of this algebra by central elements, which we call the special Racahalgebra $sR(n)$, is then introduced. Using results from classical invarianttheory, this $sR(n)$ algebra is shown to be isomorphic to the centralizer$Z_{n}(\mathfrak{sl}_2)$ of the diagonal embedding of $U(\mathfrak{sl}_2)$ in$U(\mathfrak{sl}_2)^{\otimes n}$. This leads to a first and novel presentationof the centralizer $Z_{n}(\mathfrak{sl}_2)$ in terms of generators and definingrelations. An explicit formula of its Hilbert-Poincar\'e series is alsoobtained and studied. The extension of the results to the study of the specialAskey-Wilson algebra and its higher rank generalizations is discussed.

Racah algebras, the centralizer $Z_n(\mathfrak{sl}_2)$ and its Hilbert-Poincar\'e series