Dtyjlui

I'm now dealing with this integral and trying to give a closed form to it:
$$I_n=int_0^1 prod_{l=1}^nleft[x^2-dfrac{l^2}{n^2}right]dx$$
Where the first few values are given:
$$I_1=int_0^1 (x^2-1) dx=-dfrac23$$
$$I_2=int_0^1 left(x^2-dfrac14right)(x^2-1)dx=dfrac1{30}$$
$$I_3=int_0^1 left(x^2-frac19right)left(x^2-frac49right)(x^2-1)dx=-dfrac{136}{8505}$$
I've tried to expand the polynomial, but I could merely give a general expression for the first few and the last few coefficients, and they become complicated very fast. I have searched the Internet; did I miss something that I couldn't find them because I don't know the Theorem's name?

$$I_n=frac{1}{n^{2n}}int_0^1prod_{l=1}^{n}[(xn)^2-l^2]dxqquad x=frac{y}{n}$$
$$I_n=frac{1}{n^{2n+1}}int_0^nprod_{l=1}^{n}[y^2-l^2]dy$$
We can expand the product, without knowing the coefficients for now as following;
$$prod_{l=1}^{n}[y^2-l^2]=sum_{j=0}^n(-1)^{n-j}y^{2j}t(n,j)$$
$$I_n=frac{1}{n^{2n+1}}sum_{j=0}^n(-1)^{n-j}t(n,j)int_0^ny^{2j}dy$$
$$I_n=frac{1}{n^{2n+1}}sum_{j=0}^nfrac{(-1)^{n-j}t(n,j)n^{2j+1}}{2j+1}$$
$$I_n=frac{(-1)^n}{n^{2n}}sum_{j=0}^nfrac{(-1)^{j}t(n,j)n^{2j}}{2j+1}$$
Now back to the coefficients, ive found it on OEIS A008955

they are called the central factorial numbers. I think this is the closest we can get for a closed form.

You can relate the product to the rising factorials $x^overline{n} = x(x+1)dotsb (x+n-1) = sum_{k=0}^n begin{bmatrix}n\kend{bmatrix} x^k$ where the coefficients are Stirling's numbers of the first kind.
begin{align}
I_n
&= int_0^1 prod_{l=1}^n left(x^2 - frac{l^2}{n^2}right);dx
= int_0^1 prod_{l-1 =k=0}^{n-1} frac{xn - k + 1}{n}cdot prod_{k=0}^{n-1} frac{xn + k + 1}{n};dx\
&= frac{(-1)^n }{n^{2n}}int_0^1 (-xn - 1)^{overline{n}}(xn +1)^{overline{n}};dx\
%
&= frac{(-1)^n }{n^{2n}}int_0^1 sum_{i=0}^nsum_{j=0}^n begin{bmatrix}n\iend{bmatrix}
begin{bmatrix}n\jend{bmatrix} (-xn - 1)^i (xn + 1)^j;dx\
%
&= frac{(-1)^n }{n^{2n}} sum_{i=0}^nsum_{j=0}^n (-1)^i begin{bmatrix}n\iend{bmatrix}
begin{bmatrix}n\jend{bmatrix} int_0^1 (xn + 1)^{i+j} ;dx\
%
&= frac{(-1)^n }{n^{2n}} sum_{i=0}^nsum_{j=0}^n (-1)^i begin{bmatrix}n\iend{bmatrix}
begin{bmatrix}n\jend{bmatrix} frac{(n+1)^{i+j+1}-1}{n(i+j+1)}\
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&= sum_{i=0}^nsum_{j=0}^n (-1)^{i+n}begin{bmatrix}n\iend{bmatrix}
begin{bmatrix}n\jend{bmatrix} frac{(n+1)^{i+j+1}-1}{n^{2n+1}(i+j+1)}\
%
end{align}