$$[x^n](1-x)^{-\frac{1}{2}} \exp\left(\frac{x(1+x)}{2-2x}\right)\frac{\left(\frac{2-x}{2-2x}\right)^k}{k!}\cdot\,_0 F _1 \left(;k+1;x\left(\frac{2-x}{2-2x}\right)^2\right)$$