summation of infinite series Find$$S=1+\frac{1}{6}+\frac{1}{18}+\frac{7}{324}+\cdots\infty$$
I could figure this pattern for denominators: $t_k=3t_{k-1}t_{k-2}$ but not sure how to proceed.

Let me try for Newton's generalised binomial theorem with $x=1$
So, we have $\displaystyle ry=\frac16\ \ \ \ (1)$ and $\displaystyle\frac{r(r-1)}{2!}y^2=\frac1{18}\  \ \ \ (2)$
Squaring $(1)$ and dividing by $(2)$ we get $\displaystyle r=-\frac13$ and consequently  $\displaystyle y=-\frac12$
which satisfies $$\frac{r(r-1)(r-2)}{3!}y^3=\frac7{324}$$
So, the sum will be $$\left[1+\left(-\frac12\right)\right]^{-\frac13}$$
