Recursive formula for $\psi^g-\lambda_1\psi^{g-1}+\cdots+(-1)^g\lambda_g$ in $\overline{\mathcal{M}}_{g,1}$

Mumford proved that $\psi^g-\lambda_1\psi^{g-1}+\cdots+(-1)^g\lambda_g=0$ in the Chow ring of $\overline{\mathcal{M}}_{g,1}$. We find an explicit recursive formula for $\psi^g-\lambda_1\psi^{g-1}+\cdots+ (-1)^g\lambda_g$ in the tautological ring of $\overline{\mathcal{M}} _{g,1}$ as a combination of classes supported on boundary strata.