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For a normalized positive sequence $h=(h(n))_{n\\ge 1}$ with $h(1)=1$, define $\\pol _0^h(z)=1$ and, for $n\\ge 1$, \\[ \\pol _n^h(z)=\\frac{z}{h(n)}\\sum_{k=1}^n \\sigma(k)\\pol _{n-k}^h(z),\\] where $\\sigma(k)$ denotes the sum of divisors of $k$. The Nekrasov--Okounkov polynomials are obtained from the specialization $h(n)=n$ by the shift $\\nop _n(z)=\\pol _n^h(z+1)$. We derive a Hessenberg determinant representation for $\\pol _n^h(z)$. 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