In earlier times Anton would have been given the last name Klasson because he was the son of Klas (Klas's son). A patronymic name was based on the father's first name.
But Swedes - and other Scandinavians - dropped the patronymic naming custom and adopted true surnames (a last name that passed from father to children, without regard to the father's first name).
Apparently this family settled on the surname Jacobsen. Anton came to use his father's last name as a surname (just as we're familiar with in the USA). In other words, Klas was the first dad in his line to NOT have his own first name used by his son. Instead his son used Klas's last name.
The surname Jacobson is thus traced to Jacob Petter Persson (1803 - 1851). Everybody in the generations after Jacob became "Jacob's son." This makes Jacob Petter Persson kinda special, I think. : D
Maybe you know all this - thought I'd comment in case it's confusing.