The nth Bell number Bn is defined to be the number of distinct partitions of a set of size n. The displayed formula represents the Bell number Bn+1 recursively in terms of the lower-indexed Bell numbers.
Bell numbers are named for Eric Temple Bell, a mathematician closely associated with the early history of the University of Washington mathematics department. Here is a passage about Bell from a 1991 history of the department written by Gloria Campbell:
Like the rest of campus, the mathematics department expanded in the early 20th century. Instrumental to its growth was the appointment of Professor Robert Moritz to the mathematics and astronomy faculty. ... Moritz's stay at Washington was to be a long one. For thirty-two years he remained head of the Department of Mathematics and, for many years, chaired the astronomy department as well. ...
One of the most colorful of [his] early appointments was E.T. Bell. Although he was born in Peterhead, Scotland, Bell came to the United States for his university education, completing an A.B. at Stanford, an M.A. at the University of Washington in 1908, and a Ph.D. at Columbia in 1912. The mathematics department at Washington thought highly of Bell and began negotiations to bring him back to Seattle in the summer of 1912. On September 5, Bell wired Moritz accepting his offer. Bell's lifelong mathematical interest was in elliptic functions and number theory. During his academic career, which lasted from 1912 until 1953, he produced some 200 mathematical research papers. While at the University of Washington, he published numerous significant contributions on analytic number theory, multiply periodic functions, and Diophantine analysis. In addition to his research writing, Bell published fifteen books on mathematics. Two of these - Men of Mathematics and The Development of Mathematics - are still widely read. A prodigious writer, Bell wrote seventeen science fiction novels under the pseudonym "John Taine." He also authored one short story as well as some poetry.
His productivity was well noted and, in 1922, he was promoted to full professor at an annual salary of $3,500. Within a year the Department upped Bell's salary to $4,500 to meet a competing offer. In December 1924, Bell received the Bôcher prize from the American Mathematical Society for a paper on "Arithmetrical Paraphrases." The paper was published in Transactions of the American Mathematical Society. As Bell pointed out to University President Suzzallo, the Bôcher prize was awarded every five years to the author of the most significant research published in Transactions.
Seven months later, Bell's salary was raised to $5,000 to counter a $6,000 offer from the University of Chicago. Outside offers finally prevailed, however. Bell left the University of Washington in 1926 for the California Institute of Technology where he remained until his retirement in 1953.
The early recipients of the American Mathematical Society's Bôcher Prize read like a who's who of American mathematics. It was first awarded, in 1923, to Harvard mathematician George David Birkhoff. Bell shared his award in 1924 with Solomon Lefschetz, newly arrived that year at Princeton. Next came Lefschetz's Princeton colleague James Alexander in 1928, Marston Morse and Norbert Weiner in 1933, and John von Neumann in 1938. (In 2011, Gunther Uhlmann became the second UW recipient of the Bôcher Prize.)
Bell's writing on mathematical history and his fiction may have had some features in common. Additional background on Bell can be found here. See also mathematical biographer Constance Reid's book The Search for E. T. Bell: Also Known as John Taine.