How to confirm the nth Catalan Number given the first few values in the sequence?

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Given $Y_0 = 1$, $Y_1 = 1$, $Y_2 = 2$, $Y_3 = 5$, $Y_4 = 14$, $Y_5 = 42$, $Y_6 = 132$; what are three different ways to confirm that $Y_r$ may possibly be the nth Catalan number for all $r$. Describe in detail and what are the advantages of each approach relative to the others.