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I consider polynomials $p_n(z)$ such that $p_0(z) = 1$, $p_{n+1}(z) = ( p_{n}'(z)-p_{n}(z) )z$, so $p_1(z) = -z$, $p_2(z) = z(z-1)$, $p_3(z) = -z + 3z^2 - z^3$. Are they well-known? Do they have their own name?

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I think the coefficients of your polynomials are Stirling numbers of the second kind. Lots of information here. In particular, the polynomials seem to be the moments of a random variable with a Poisson distribution.

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Thank you. Polynomial with Stirling numbers coefficients is known as Bell polynomial. mathworld.wolfram.com/BellPolynomial.html –  Nimza Jan 30 '12 at 10:30
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