Enumerating functions for nonnegative integer coordinates of $L$-dimensional vectors

The enumerating function $C^L(X_1,\dots,X_L)$, which bijectively maps tuples of length $L$ of nonnegative integers to nonnegative integers $Z=C^L(X_1,\dots,X_L)$, is represented as a sum of $L$ figurate numbers.