Let $n$ integer $\geq 2$ (or big enough) fixed and for $x$ real, let $$P_n(x)=\displaystyle \sum_{k=0}^{n-1} C_{n+k}^n (-x)^k \alpha_{n,k}$$ and where $\forall k$ such that $0 \leq k \leq n-1 $ $$ \alpha_{n,k}= \displaystyle \sum_{p=1}^{n-k} \displaystyle C_{n}^{n-k-p} \frac{(-1)^{p+1}}{p}$$.

Numerically I have found the following: $ \forall 2 \leq n \leq 10 $ ,

$$\max_{[0,1]} |P_n(x)|=|P_n(0)|=\sum_{j=1}^{n} \frac{1}{j}.$$

Is there an article about $P_n$ or similar polynomials?

I need a proof that it's true $\forall n \geq 2 $.

I have noticed numerically too that $P_n(x)=(-1)^{n+1}P_{n}(1-x)$ and $P_n$ have exactly $n-1$ zeros over $]0,1[$. All this properties are very similar to those of Legendre polynomial: $L_n(x)=\displaystyle \frac{1}{n!}(x^n(1-x)^n)^{(n)}$.

I suppose that the family of $P_n$ are orthogonal but I don't know to what, and maybe satisfy a recurrence relation...

If anyone suggests an idea for (the proof that I suppose true) thanks for his help.

I'm looking too for an integral representation of type $ \int_{0}^1 f^n(t).g(t)dt$ for $P_n(x), 0<x<1$ but it's not very important for me, what interests me mostly is the maximum of $P_n$ over $[0,1]$.