For every n, the identity sum_{k=0}^n 2^{n-k} s(n,k) B_k = sum_{k=0}^n b(n,k) (-1)^k k!/(k+1) is proved.
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A New Identity Linking Bernoulli Numbers, Stirling Numbers of the First Kind, and Bessel Numbers of the First Kind
For every n, the identity sum_{k=0}^n 2^{n-k} s(n,k) B_k = sum_{k=0}^n b(n,k) (-1)^k k!/(k+1) is proved.