He was born in
Bucharest. His father, also named Traian, was originally from
Cornea, Caraș-Severin and worked as a superintendent at the Creditul Agricol Bank. Lalescu went to the
Carol I High School in
Craiova, continuing high school in
Roman, and graduating from the
Boarding High School in
Iași. After entering the
University of Iași, he completed his undergraduate studies in 1903 at the
University of Bucharest. He earned his
Ph.D. in Mathematics from the
University of Paris in 1908. His dissertation,
Sur les équations de Volterra, was written under the direction of
Émile Picard. That same year, he presented his work at the
International Congress of Mathematicians in Rome. A year later, he was appointed full-time professor of analytical geometry, succeeding
Spiru Haret; he lectured at the School (which would later become the
Polytechnic University of Bucharest) until his death. In 1916, he became the first president of
Sportul Studențesc, the university's
football club. Also that year, he was appointed tenured professor of algebra and number theory at the
University of Bucharest, a position he held until his death. One of his PhD students there was
Valeriu Alaci. In 1920, Lalescu became a professor and the inaugural
rector of the
Polytechnic University of Timișoara; for a year, he would commute by train for 20 hours between Timișoara and Bucharest to teach his classes. ==The Lalescu sequence==