Sums of sixteen and twenty-four triangular numbers (Q2566526)

Let \(\delta_k(n)\) be the number of representations of \(n\) as the sum of \(k\) triangular numbers, which are of the form \(\frac {m(m+1)}2\) for \(m=0,1,2,\dots\)\,. The authors give elementary proofs of the Kac-Wakimoto formulas for \(\delta_{16}(n)\) and \(\delta_{24}(n)\) [\textit{V. G. Kac} and \textit{M. Wakimoto}, Prog. Math. 123, 415--456 (1994; Zbl 0854.17028)]. For the proof, they derived suitable identities involving convolution sums of divisor functions from an elementary arithmetic identity proved by the authors in a joint work with \textit{Z. M. Ou} and \textit{B. K. Spearman} [Number theory for the millennium II, 229--274 (2002; Zbl 1062.11005)].