Solution regarding Power Series and ODE's

Question

About 4 months ago I posted Series solution to $y''-xy'-y=0$. I ran through the analysis and it appeared that I solved the ODE . The solution seemed to be $$y=a_0+\frac{a_1}{1}x+\frac{a_0}{2}x^2+\frac{a_1}{1 \cdot 3}x^3+\frac{a_0}{2 \cdot 4}x^4+\frac{a_1}{1 \cdot 3 \cdot 5}+...$$ $$=a_0\sum_{k=0}^{\infty}\frac{x^{2k}}{2^kk!}+a_1\sum_{k=0}^{\infty}\frac{2^kk!x^{2k+1}}{(2k+1)!}$$ It appears that my summation is wrong and, at least, the $a_0$ term should be $$\frac{a_0}{2}\sum_{k=0}^\infty\frac{x^{2k}}{k!}$$ since my $a_0$ terms are $$\frac{a_0x^{2\cdot 0}}{2(1)},\frac{a_0x^{2\cdot 1}}{2(1)},\frac{a_0x^{2\cdot 2}}{2(2\cdot 1)},\frac{a_0x^{2\cdot 3}}{2(3\cdot 2\cdot 1)},\frac{a_0x^{2\cdot 4}}{2(4\cdot 3\cdot 2\cdot 1)},...$$ Thus, one possible solution seems to be $y=\frac{e^{x^2}}{2}$, since $$e^{x^2}=\sum_{k=0}^{\infty}\frac{(x^2)^k}{k!}.$$ But when I differentiate $y$ twice and plug the derivitives and original equation in the ODE, I get $$y''-xy'-y=0$$ $$2x^2e^{x^2}+e^{x^2}-x^2e^{x^2}-\frac{e^{x^2}}{2}\neq 0$$ So.....thoughts? Why am I not calculating right? Am I just missing something simple?