The author gives a purely combinatorial proof of the identity \[ \prod_{n>m}(1-q^n) = \sum_{n=1}^{\infty} (-1)^n \left[ {n+m \atop m} \right] q^{nm+n(3n+1)/2}(1-q^{2n+m+1}), \] which generalizes Franklin's proof for the case \(m=0\).