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Let $B_n$ be the $n$-th Catalan Number. We have $ B(x) = \sum_{n \ge 0} B_n x^n = \frac{1-\sqrt{1-4x}}{2x}$.

Does anyone know a closed form of the generating function of the shifted Catalan Numbers, i.e. for chosen $p_0$, for $B_{p_0}(x) = \sum_{n \ge 0} B_{n+p_0} x^n$?

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    Curious, what's the point of the subscript $0$?2012-02-10
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    Hm, you're right, I could have skipped that. I had the subscript in my notes for a different reason and forgot to take it out for the post.2012-02-10
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    The rational function 1/(1-x/(1-x/(1-x/(1- ... x/1)))) with n x's agrees with B(x) modulo x^{n+1}. Does that help? (Probably not, but maybe.)2012-06-05
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    Well, that's a very interesting result, how do I prove it/where can I find a proof? But I can't yet see how it could help me, because I don't want to take my function mod anything. The real question is linked (asymptotic of shifted Catalan Numbers), I still haven't solved it and I'd be grateful for any help.2012-06-05

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