I am trying to create a sequence of polynomials,
$$s_n = \{x^0+x,\quad x^1+x,\quad x^2+x,\quad ...,\quad x^n+x\}$$
Using recursion.
That is, I want to "convert" $f(n) = x^n + x$ into a recursive function g(n). For example, I've found that $$ \begin{aligned} &f(n) = 2x^n + x = g(n,c), \quad n=c\\ &g(n,c) = x\cdot g(n-1,c) + \frac{1}{(c-1)x^{c-n-1}}\\ &g(0,c) = x+1,\quad g(1,c) = 2x\\ &n, c \quad\text{ (are natural numbers)}\\ &\text{e.g. } f(5) = g(5,5) \end{aligned} $$
But I don't know how to get a similar recursive function for $x^n+x$.
How about $s_{n+1}= x(s_n-x)+x$?