On the Representation of Integers as Sums of a Class of Triangular Numbers

In this paper, we discuss the problem of the number of representations of positive integers as sums of triangular numbers. The method we use is similar to Rankin's way in studying the sum of squares representation of positive integers. We decompose the theta function $q^{k}\psi ^{4k}(q)\psi ^{2k}({q^2})$ into an Eisenstein series and a cusp form to give an asymptotic formula for $t_{4k,2k}(n)$. Moreover, we obtain concrete formulas for $k = 2,4$, respectively, by using a linear combination of the divisor function and the coefficient of an $\eta$-product.