I first got interested in genealogy when I heard that my Granny Jackson was from the same Jackson family in Northern Ireland that produced President Jackson. In the last couple of years I have taken up researching my Jackson line.
Unfortunately, it may be that not much will ever be known about President Jackson's paternal family. He lost his father while a toddler and his mother while a teenager, so presumably all he knew about his Jacksons was filtered through his aging memory of whatever his mother had told him and his brothers. Jackson himself showed no interest in connecting with cousins, he only wrote one very brief account of his Jackson side, and almost none of it has ever been substantiated through primary records (in spite of significant searches for said documents). So little is known, even the names of Jackson's paternal uncles and grandfather are disputed. There are currently multiple profiles on Wikitree claiming to be his Jackson grandfather!
But here's why I don't really care that my family lore will probably never be confirmed or disproved. I found the ancestral Irish townland for my Jacksons in 2017, and with it I found a bunch of cousins with paper trails and matching DNA segments. We're now a big bunch of penpals, and it's great fun. And before we found each other, most of us had concluded we would probably never find Jackson cousins.
But maybe the most fun I'm having is by using my family lore as genealogical evidence! See, a man named Robert Jackson wrote from Pennsylvania in 1822 to then General Jackson, claiming to be the general's uncle. The letter was saved and is in Jackson's correspondence, but unfortunately, Jackson never gave his opinion about the contents of the letter. It does have some tantalizing commonalities with his own account: Robert says he was one of four brothers and that Andrew Jackson's older brothers Hugh and Robert were born in Ireland and at sea, respectively. He also writes that he had received three letters from Andrew Jackson's father before he passed.
So I started researching Robert Jackson, and I found one big problem with his account: evidence indicates that Robert was from County Tyrone, and no document I have seen or heard of connects President Jackson to Tyrone. But guess what: my own line comes from Tyrone! Lots of families claim descent from Jackson's grandfather, but families from Tyrone claiming kinship with President Jackson are very rare!
So I have been working on trying to prove that my Jacksons and Robert were related. For instance, in the records of Carland Church, to which my Jacksons belonged, there is the birth of a Hugh Jackson to Andrew Jackson in 1760. The father Andrew disappears from the records in the mid 1760's, at the same time that President Jackson's father emigrated. This could very well be the Hugh Jackson who Robert believed was President Jackson's older brother.
In the Carland records is also the birth of Andrew Jackson to Robert Jackson 1765, and Pennsylvania Robert had a son Andrew born in the mid 1760's.
I know of a few Jackson men in my line, but none responded to my inquiry to see if they had interest in y-chromosome testing. And I probably won't pursue that further until I find a Jackson descendant of Robert.
I guess I'll wrap up by saying that President Jackson himself had no known biological offspring. Lots of people can confirm relationship to President Jackson's mother, Elizabeth Hutchinson Jackson, or to his wife Rachel Donelson Jackson, but as far as I can tell, no one alive can confirm a connection on Jackson's paternal side. (There are claims that Jackson's y-chromosome is known, but such claims seem to be based on the supposition that people know who Jackson's paternal uncles or grandfather was. And solid proof just isn't there.)