Hey guys! I'm doing an assignment, and I'm just not sure (at all) how to start this problem. Can somebody nudge/shove me in the right directions?

Show that the Catalan numbers are given by the recurrence relation

(n+2)C$_{n+1}$ = (4n+2)C$_n$

and initial condition C$_0$ = 1

Thanks in advance!

  • 1
    $\begingroup$ Hint: start with the definition of the Catalan numbers, written out in terms of factorials. Plug this in to the recurrence, divide the left side by the right side, and see if you can simplify the result to 1. $\endgroup$ – Robert Israel Apr 29 '11 at 5:55
  • $\begingroup$ Great! That worked out nicely. Sort of had a facepalm moment after reading that. Thank you! $\endgroup$ – Arthur Skirvin Apr 29 '11 at 6:27

This hint comes in two parts:

I will assume that you know that $$C_n = \frac{1}{(n+1)!} \prod_{k=1}^n (4k - 2)$$

Secondly, what is $\dfrac{C_{n+1}}{C_n}$ ?

  • $\begingroup$ As happens from time to time, I posted this the same time that Robert Israel posted his comment - and so he deserves equal pride in the answer. $\endgroup$ – davidlowryduda Apr 29 '11 at 5:58
  • $\begingroup$ Equal thanks to you! $\endgroup$ – Arthur Skirvin Apr 29 '11 at 6:29

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