100 and more conjectures from the OEIS/Proof 1

First, we put the expression $(1-4x{)}^{3/2}$ into the form $(1-4x)\sqrt{1-4x}=\sqrt{1-4x}-4x\sqrt{1-4x}$. Now, using the binomial theorem we obtain User:Rwst/Eq Likewise, $[x_n]4x\sqrt{1-4x}=4\frac{(\genfrac{}{}{0ex}{}{2n-2}{n-1})}{3-2n}$, so we just need to manipulate factorials to get the desired result:

$\begin{array}{cc}[x_n](1-4x{)}^{3/2}& =\frac{(\genfrac{}{}{0ex}{}{2n}{n})}{1-2n}-4\frac{(\genfrac{}{}{0ex}{}{2n-2}{n-1})}{3-2n}\\ & =\frac{4n^2(2n-1)(2n-2)!-(2n-3)(2n)!}{(2n-1)(2n-3)n{!}^2}\\ & =\frac{4n^2(2n-1)(2n-2)(2n-4)!-(2n)!}{(2n-1)n{!}^2}\\ & =\frac{4n^2(2n-2)(2n-4)!-2n(2n-2)!}{n{!}^2}\\ & =\frac{(2n-4)![4n^2(2n-2)-2n(2n-2)(2n-3)]}{n!n(n-1)(n-2)!}\end{array}$