Very frequent multiple births in humans
Risk factors for dizygotic twinning are related to multiple follicular development, and include maternal family history, ethnicity, geography, maternal parity, maternal age, and, of course, use of assisted reproductive technology. There may be a genetic component to monozygotic twinning as well, but that rate is fairly consistent across populations. Other risk factors, such as diet and supplementation, have been proposed, but the data are less robust. There are some interesting studies demonstrating geographic clusters of twinning, but these tend to be demographic clusters that are associated with other risk factors. 

Though a higher risk of twinning can be transmitted from a father to his daughters, the father's family history of twinning is not a significant risk factor (for his own children to be twins). To clarify -- if a man has a family history of multiple births, his children are no more likely to be twins than the general population, but his daughers are more likely to give birth to twins. This study is one example of the studies that have shown a significant independent association between maternal family history and twinning, with no significant independent association with paternal family history.

As far as the case of Feodor Vassilyev is concerned, many people have lived on this earth, making very unlikely events in a single instance more likely to have ever occurred. I would say this case is either a hoax or a rare chance event. There is no modern data I'm aware of that would support a paternal genetic reason that this particular man produced so many multiples.