Expansion of the R1a1a Ashkenazi Levite Population

As of the start of the 15th century, there were only about 50,000 Ashkenazi Jews; by the start of the 19th century, there were about 5,000,000 Ashkenazi Jews. See G. Atzmon et al., Abraham’s Children in the Genome Era: Major Jewish Diaspora Populations Comprise Distinct Genetic Clusters with Shared Middle Eastern Ancestry, Am. J. Hum. Gen. 86:850 (2010). This is a hundredfold increase.

Assuming for ease of analysis that the proportion of R1a1a Ashkenazi Levites in the Ashkenazi population at the start of the 15th and 19th centuries was the same as the current population estimated in the 2003 paper by Behar et al. (approximately 2% of the total population), that influx from Sephardi and Mizrahi populations did not increase the Ashkenazi population between the start of the 15th century and the start of the 19th century, and that the R1a1a Ashkenazi Levite progenitor living at the start of the 15th century was the ancestor of 50% of R1a1a Ashkenazi Levites (as suggested by initial STR analyses posted on this website), that man would have had about 25,500 direct male descendants (and 50,000 descendants born to a father on his direct male line) as of the start of the 19th century. 

Assuming for ease of analysis that the proportion of R1a1a Ashkenazi Levites in the Ashkenazi population at the start of the 15th and 19th centuries was the same as the current population estimated in the 2013 paper by Rootsi & Behar et al. (approximately 13% of the total population), that influx from Sephardi and Mizrahi populations did not increase the Ashkenazi population between the start of the 15th century and the start of the 19th century, and that the R1a1a Ashkenazi Levite progenitor living at the start of the 15th century was the ancestor of 20% of R1a1a Ashkenazi Levites (as suggested by more recent STR and SNP analyses posted on this website), that man would have had about 65,000 direct male descendants (and 130,000 descendants born to a father on his direct male line) as of the start of the 19th century. 

Such reproductive success is mathematically feasible, assuming that in the early generations R1a1a Ashkenazi Levite men had a substantial number of sons who in turn had a substantial number of sons.
Mordecai Aronzon & Family

Mordecai Aronzon (1882-1920), Shava
Rubenstein (1882-1945) & children
Photograph taken in @1917 in or near
Verbovets, Podolia, Ukraine