The author provides an easy proof of the recurrence
where are the Bernoulli numbers. The author uses this formula to present proofs of theorems on Bernoulli numbers due to Staudt-Clausen, Carlitz, Frobenius and Ramanujan. An analogous recurrence for Genocchi numbers is given, which is used to present new proofs of theorems on Genocchi numbers of Lehmer, Ramanujan and Kummer. In some cases, earlier results are extended.