When the Sun eventually dies out will a new one be formed

We all know very well that the Sun will burn out in 5 billion years time, But when it actually does happen will a new one eventually form again?

Some of the matter ejected by the Sun when the core collapses and it becomes a white dwarf at the center of a planetary nebula, most certainly will, at some point in the distant future, become part of another star. This ejected matter would have to collide with enough dust and particles from the interstellar medium, to become dense enough to ever do so. Depending on how the Andromeda-Milky Way merger pans out in a few billion years, this might mean there could be an abundance of opportunities for this to happen, it could also mean our Sun might be ejected from galactic orbit, and no new star will form... probably somewhere in between these two extremes, lies the reality.

Once we get out of the main sequence phase of the life of the sun it will start to fuse helium. Then it will expand and become a red giant for a while.

Once the helium runs out the sun will throw off its outer layers and form a planetary nebula. This is not enough material to form another star. It will then shrink to a white dwarf and will cool off slowly over the course of several billion years.

The time to cool down completely is estimated to be longer than the Universe has been in existence. Say longer than 13 Billion years.

For a smaller star to be formed out of the ashes of another dying star, there needs to be enough matter to do that. You only get that in the explosions of supermassive stars.

Ah this I can answer Wizard!!!!

Sadly, no.

The sun will change into a red giant 5 bil yrs from now (It'll grow cooler but larger; with the overall effect that it will burn the Earth because the distance between the sun and earth will reduce). It will change into one because it will burrn all its hydrogen and then will have to start burning Helium (which is heavier)...