How do I notate a polynomial with Stirling coefficients and what properties do I need to prove it?

Question

I have a group of polynomials where each term increases in degree and has coefficients that appear as Stirling numbers of the second kind:

$1: 1 \\ 2: 1+x \\ 3: 1+3x+x^2 \\ 4: 1+7x+6x^2+x^3 \\ 5: 1+15x+25x^2+10x^3+x^4 ...\\$

How can I represent consecutive polynomials in closed form and what properties of Stirling numbers and/or polynomials do I need to prove the nth polynomial with induction?

Answer 1

Hint: These polynomials are called Touchard polynomials \begin{align*} T_n(x)=\sum_{k=0}{n \brace k}x^k\qquad n\geq 0 \end{align*} and you might want to use \begin{align*} T_{n+1}(x)=x\sum_{k=0}^n\binom{n}{k}T_k(x) \end{align*} for induction.