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This question is inspired by the MO query here, although it has no direct implications.

Define the family of polynomial functions $$f_n(x)=n^2x^{n-1}-\frac{d}{dx}\left(\frac{x^n-1}{x-1}\right),$$ and the associated family of algebraic functions $$g_n(x)=\frac{\sqrt{f_n(x)}}n+\sum_{j=1}^{n-1}\frac{\sqrt{f_j(x)}}{j(j+1)}.$$

Question. Despite the complicated expressions for $f_n$ and $g_n$, does $$h_n(x)=\sum_{k=0}^n\left(g_n(x)-g_k(x)\right)^2$$ have a neat (closed) formula?

UPDATE. user64494 (see below) has found $h_n(x)=nx^{n-1}$. Any proof?

Convention: $g_0:=0$ and empty sums are zero.

For example, $h_2(x)=g_2(x)^2+(g_2(x)-g_1(x))^2+(g_2(x)-g_2(x))^2=2x$.

This question is inspired by the MO query here, although it has no direct implications.

Define the family of polynomial functions $$f_n(x)=n^2x^{n-1}-\frac{d}{dx}\left(\frac{x^n-1}{x-1}\right),$$ and the associated family of algebraic functions $$g_n(x)=\frac{\sqrt{f_n(x)}}n+\sum_{j=1}^{n-1}\frac{\sqrt{f_j(x)}}{j(j+1)}.$$

Question. Despite the complicated expressions for $f_n$ and $g_n$, does $$h_n(x)=\sum_{k=0}^n\left(g_n(x)-g_k(x)\right)^2$$ have a neat (closed) formula?

UPDATE. user64494 (see below) has found $h_n(x)=nx^{n-1}$. Any proof?

Convention: $g_0:=0$ and empty sums are zero.

For example, $h_2(x)=g_2(x)^2+(g_2(x)-g_1(x))^2+(g_2(x)-g_2(x))^2=2x$.

This question is inspired by the MO query here, although it has no direct implications.

Define the family of polynomial functions $$f_n(x)=n^2x^{n-1}-\frac{d}{dx}\left(\frac{x^n-1}{x-1}\right),$$ and the associated family of algebraic functions $$g_n(x)=\frac{\sqrt{f_n(x)}}n+\sum_{j=1}^{n-1}\frac{\sqrt{f_j(x)}}{j(j+1)}.$$

Question. Despite the complicated expressions for $f_n$ and $g_n$, does $$h_n(x)=\sum_{k=0}^n\left(g_n(x)-g_k(x)\right)^2$$ have a neat (closed) formula?

UPDATE. user64494 (see below) has found $h_n(x)=nx^{n-1}$. Any proof?

Convention: $g_0:=0$ and empty sums are zero.

Example. $h_2(x)=g_2(x)^2+(g_2(x)-g_1(x))^2+(g_2(x)-g_2(x))^2=2x$.

This question is inspired by the MO query here, although it has no direct implications.

Define the family of polynomial functions $$f_n(x)=n^2x^{n-1}-\frac{d}{dx}\left(\frac{x^n-1}{x-1}\right),$$ and the associated family of algebraic functions $$g_n(x)=\frac{\sqrt{f_n(x)}}n+\sum_{j=1}^{n-1}\frac{\sqrt{f_j(x)}}{j(j+1)}.$$

Question. Despite the complicated expressions for $f_n$ and $g_n$, does $$h_n(x)=\sum_{k=0}^n\left(g_n(x)-g_k(x)\right)^2$$ have a neat (closed) formula?

UPDATE. user64494 (see below) has found $h_n(x)=nx^{n-1}$. Any proof?

Convention: $g_0:=0$ and empty sums are zero.

For example, $h_2(x)=g_2(x)^2+(g_2(x)-g_1(x))^2+(g_2(x)-g_2(x))^2=2x$.

This question is inspired by the MO query here, although it has no direct implications.

Define the family of polynomial functions $$f_n(x)=n^2x^{n-1}-\frac{d}{dx}\left(\frac{x^n-1}{x-1}\right),$$ and the associated family of algebraic functions $$g_n(x)=\frac{\sqrt{f_n(x)}}n+\sum_{j=1}^{n-1}\frac{\sqrt{f_j(x)}}{j(j+1)}.$$

Question. Despite the complicated expressions for $f_n$ and $g_n$, does $$h_n(x)=\sum_{k=0}^n\left(g_n(x)-g_k(x)\right)^2$$ have a neat (closed) formula?

Convention: $g_0:=0$ and empty sums are zero.

Example. $h_2(x)=g_2(x)^2+(g_2(x)-g_1(x))^2+(g_2(x)-g_2(x))^2=2x$.

This question is inspired by the MO query here, although it has no direct implications.

Define the family of polynomial functions $$f_n(x)=n^2x^{n-1}-\frac{d}{dx}\left(\frac{x^n-1}{x-1}\right),$$ and the associated family of algebraic functions $$g_n(x)=\frac{\sqrt{f_n(x)}}n+\sum_{j=1}^{n-1}\frac{\sqrt{f_j(x)}}{j(j+1)}.$$

Question. Despite the complicated expressions for $f_n$ and $g_n$, does $$h_n(x)=\sum_{k=0}^n\left(g_n(x)-g_k(x)\right)^2$$ have a neat (closed) formula?

UPDATE. user64494 (see below) has found $h_n(x)=nx^{n-1}$. Any proof?

Convention: $g_0:=0$ and empty sums are zero.

Example. $h_2(x)=g_2(x)^2+(g_2(x)-g_1(x))^2+(g_2(x)-g_2(x))^2=2x$.