Journal of Integer Sequences, Vol. 22 (2019), Article 19.7.1

On Engel's Inequality for Bell Numbers

Horst Alzer
Morsbacher Straße 10
51545 Waldbröl


We prove that for all integers n ≥ 2 the expression Bn-1 Bn+1 - Bn2 can be represented as an infinite series with nonnegative terms. Here Bk denotes the k-th Bell number. It follows that the sequence (Bn)n ≥ 0 is strictly log-convex. This result refines Engel's inequality Bn2Bn-1 Bn+1.

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(Concerned with sequence A000110.)

Received June 26 2019; revised version received August 25 2019. Published in Journal of Integer Sequences, September 25 2019.

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