Tritriangular Number

A number of the form

(Comtet 1974, Stanley 1999), where  is a binomial coefficient. The first few values are 3, 15, 45, 105, 210, 378, 630, ... (OEIS A050534). The generating function for the tritriangular numbers is

Given  lines in a plane, no two of which are parallel, and no three of which are concurrent, draw lines pairwise through their points of intersection. The number of new lines drawn is then  (Schmall 1915).

REFERENCES:

Comtet, L. Problem 1. Advanced Combinatorics: The Art of Finite and Infinite Expansions, rev. enl. ed. Dordrecht, Netherlands: Reidel, p. 72, 1974.

Schmall, C. N. "Problem 432." Amer. Math. Monthly 22, 130, 1915.

Sloane, N. J. A. Sequence A050534 in "The On-Line Encyclopedia of Integer Sequences."

Stanley, R. P. Problem 5.5, Case 2 in Enumerative Combinatorics, Vol. 2. Cambridge, England: Cambridge University Press, 1999.