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Is there another way to determine monotonicity besides $U_{n+1} - U_n$?

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    Thanks hehe :P Duly edited2010-10-30

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I'll write something so this question doesn't remain unanswered.

Sometimes it can be quite difficult to prove that a sequence is increasing. For example, Smetaniuk gave a proof that the number of Latin squares $L_n$ (Sloane's A002860) is increasing (actually, he proved $L_{n+1} \geq (n+1)!L_n$), which is one of the best results around regarding the mysterious $L_n$.

Smetaniuk, Bohdan A new construction of Latin squares. II. The number of Latin squares is strictly increasing. Ars Combin. 14 (1982), 131–145.