Is there any way to create (a closed form for) this power series/generating function?

Question

There is a fairly simple pattern to it.

$$1 y + $$

$$(1 + 1x)y^2+ $$

$$(1+1x+1x^2 + 1x^3)y^3 + $$

$$(1+1x+\dots+1x^7)y^4 + $$

$$(1+1x+\dots+1x^{15})y^5 + $$

$$\dots$$

Does anyone know of a way to get the closed form for this? In other words, a closed form for

$$1 + x + x^2 + x^3 + \dots$$

is

$$\frac{1}{1-x}$$