I wish to compute the expansion coefficient of a wavefunction(i.e. in quantum mechanics) which in itself is an integral, given by
$$b\_{n00} = \int^{\infty}\_{u = 0} \frac{1}{(n)^{3/2}} L^{1}\_{n-1}\left(\frac{2u}{n}\right)e^{-\frac{u}{n}}e^{-\frac{u^2}{4}}u^2 du$$

I've been computing it in Mathematica as follows:

```
Monitor[For[n = 1, n < 10000, n++,Print[N[Integrate[(n^(-3/2))*LaguerreL[n - 1, 1, u/n*2]*Exp[-u/n]*Exp[(-u^2)/4]*u^2, {u, 0, \[Infinity]}]]]], n] 

```

I used Monitor because I let this integral run overnight and I want to know what n-th value it evaluated to.
Now in quantum mechanics it is know that the sum of square of the expansion coefficient must equal to one, such that:
$$\sum\_{n=0}^{\infty}|b\_{n00}|^2 = 1$$
and I would like to check that with Mathematica. I tried using NSum(integral^2) in the for loop but I have no idea how to make it work. So how can I write my code so that my loop prints out the n-th value, the evaluated integral value at that particular n, and also the cumulative sum of the square of all the n-th value of the integral at a particular n?