A common misconception, that US citizenship of one parent is in and of itself sufficient to convey citizenship to offspring. Young women who have married foreign nationals and moved to the husband's county have found out the hard way that there are further requirements.
There are more than a few children trapped in ME countries, as well as some European ones (notably Germany). State won't lift a finger to help the mothers gain custody of their kids because the kids are not US citizens.
AS long as the U.S. citizen parent has lived 5 years in the U.S. (2 after the age of 15) AND can prove it, then the kid can lay claim to U.S. citizenship. I know, I’ve done it.