Sums of triangular numbers from the Frobenius determinant

Details Sums of triangular numbers from the Frobenius determinant Sums of triangular numbers from the Frobenius determinant

2005-04-13

arxiv.org/abs/math/0504272v2

Adv. Math. 208 (2007), 935-961

Mathematics

Number Theory

27 pages; minor changes from previous version

Scientific paper

We show that the denominator formula for the strange series of affine superalgebras, conjectured by Kac and Wakimoto and proved by Zagier, follows from a classical determinant evaluation of Frobenius. As a limit case, we obtain exact formulas for the number of representations of an arbitrary number as a sum of 4m^2/d triangles, whenever d divides 2m, and 4m(m+1)/d triangles, when d divides 2m or d divides 2m+2. This extends recent results of Getz and Mahlburg, Milne, and Zagier.

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