Thank you for sharing the family of John and Barbara Catherine (Franks) Barrickman. As you can see, it's been years since I posted this message and two researchers who invested years on this puzzle have now passed away. They would be thrilled to read your message.
Thank you, also, for identifying the earlier generations ... namely: Johannes Barrickman, b 1685 Christopher Barrickman, b 1712 Henry Barrickman [Sr], b 1750
I like your theory that William Barrickman, first husband of Mary Beeson, could be the missing child in the John Barrickman household. However, I suspect that Mary Beeson, b 1778, married her first husband around 1800, suggesting a Barrickman male contemporary to John Barrickman, b 1775.
In 1819, William Fulton (and wife Mrs. Mary (Beeson) Barrickman) are taxed at Perry Co, OH. Along with Jonathan Franks and others.
I'll focus on Henry Barrickman, b 1750. Thank you!