On Miki's identity for Bernoulli numbers

We give a short proof of Miki's identity for Bernoulli numbers,∑i=2n-2βiβn-i-∑i=2n-2niβiβn-i=2Hnβnfor n⩾4 where, βi=Bi/i, Bi is the ith Bernoulli number, and Hn=1+1/2+⋯+1/n.