$<a,b,c,d,e,f...>$ = $a + bx + cx^{2} + dx^{3} + ex^{4} + fx^{5} + ...$
$<1,2,1,4,1,6...>$ = $<1,1,1,1,1,1...> + <0,1,0,3,0,5,...>$
$<1,2,1,4,1,6...>$ = $\frac{1}{1-x} + <0,1,0,3,0,5,...>$
$<1,3,5,7,9,11...>$ = $<1,1,1,1,1,1,...> + <0,2,4,6,8,10...>$
$<1,3,5,7,9,11...>$ = $\frac{1}{1-x} + <0,2,4,6,8,10...>$
$<2,4,6,8,10,12...>$ = $2<1,2,3,4,5,6...>$
$<1,2,3,4,5,6...>$ = $\frac{\mathrm{d} }{\mathrm{d} x} <1,1,1,1,1,1,...>$ =$\frac{\mathrm{d} }{\mathrm{d} x}$ $\frac{1}{(1-x)}$ = $\frac{1}{(1-x)^{2}}$
$<2,4,6,8,10...>$ = $2<1,2,3,4,5,6...>$ = $\frac{2}{(1-x)^{2}}$
$<0,2,4,6,8,10...>$ = $\frac{2x}{(1-x)^{2}}$ (Multiply by $x$ on both sides of the above equation)
$<1,3,5,7,9,11...>$ = $\frac{2x}{(1-x)^{2}} + \frac{1}{1-x}$ = $\frac{2x}{(1-x)^{2}} + \frac{1-x}{(1-x)^{2}}$ = $\frac{1+x}{(1-x)^{2}}$
$<1,0,3,0,5,0,7,...>$ = $\frac{1+x^{2}}{(1-x^{2})^{2}}$ (Substitute $x$ with $x^{2}$ on both sides of the above equation)
$<0,1,0,3,0,5,...>$ = $\frac{x(1+x^{2})}{(1-x^{2})^{2}}$ (Multiply by $x$ on both sides of the above equation)
$<1,2,1,4,1,6...>$ = $\frac{x(1+x^{2})}{(1-x^{2})^{2}} + \frac{1}{1-x} $
Refer generating functions from Oscar Levin to become good at manipulating sequences. https://discrete.openmathbooks.org/pdfs/dmoi-tablet.pdf