I was reading a book about Catalan Numbers (Thomas Koshy Catalan numbers with applications)
And I was reading through that example.
Find the number of n-element multisets $\{a_1 ,a _2 , . . . , a_n \}$ of elements $a_i \in Z_{n+1}$ such that $a_1 + a_2 + . . . + a_n = 0$, the additive identity of $Z_{n+1}$ .
So it is believed this example generates the Catalan numbers sequence.
But I disagree with the table he provides .
So when $n=1$ we have that $Z_{n+1} = Z_{2}$ and so we should have that $1 + 1 = 0 + 0 = 0$ and so the count should be 2 not 1
and for the case where $n=0$ which is not inlcuded in the above table, it should be $1$ because we only have $0 + 0 = 0$
and so the sequence should be $$1,2,2,5,14,...$$
Did I do something wrong here ?