The first several are:

$$0! \cdot h_0(x) = 1$$ $$1! \cdot h_1(x) = x+1$$ $$2! \cdot h_2(x) = x^2+3 x+2$$ $$3! \cdot h_3(x) = x^3+7 x^2+12 x+6$$ $$4! \cdot h_4(x) = x^4+15 x^3+50 x^2+60 x+24$$

Feeding the sequence $2,3,1,6,12,7,1,24,60$ into the OEIS gives the following page, which contains generating functions, relations, and citations to occurrences of this sequence of polynomials in the literature.

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The first several are:

$$0! \cdot h_0(x) = 1$$ $$1! \cdot h_1(x) = x+1$$ $$2! \cdot h_2(x) = x^2+3 x+2$$ $$3! \cdot h_3(x) = x^3+7 x^2+12 x+6$$ $$4! \cdot h_4(x) = x^4+15 x^3+50 x^2+60 x+24$$

Feeding the sequence $2,3,1,6,12,7,1,24,60$ into the OEIS gives the following page, which contains generating functions, relations, and citations to occurrences of this sequence of polynomials in the literature.