co.combinatorics - Trying to sum a series (related to catalan numbers perhaps)

edit: the preceding answer suggests that my browser didn't display the dots, i.e. you really meant the series, not the sequence... Sorry, the below doesn't answer the question.

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Does the following look right? (might this be homework?)

(1) -> f n == reduce(+, [binomial(2*i, i)*2^i/3^(2*i) for i in 0..n])
                                                                   Type: Void
(2) -> guess([f n for n in 0..20], maxLevel==2)
   Compiling function f with type NonNegativeInteger -> Fraction(
      Integer)

                  s   - 1
                   21      8p   + 12
                   ++-++     20
                 4  | |    ---------
                    | |    9p   + 18
         n - 1    p  = 0     20
          --+      20
   (2)  [ >      ------------------- + 1]
          --+             9
         s  = 0
          21
                                              Type: List(Expression(Integer))
(3) -> guessPRec [f n for n in 0..20]

   (3)
   [
     [f(n): (9n + 18)f(n + 2) + (- 17n - 30)f(n + 1) + (8n + 12)f(n)= 0,
                     13
      f(0)= 1, f(1)= --]
                      9
     ]
                                              Type: List(Expression(Integer))

In general, it's often a good idea to generalise, i.e., introduce more parameters:

(4) -> f n == reduce(+, [binomial(2*i, i)*x^i/y^(2*i) for i in 0..n])
   Compiled code for f has been cleared.
   1 old definition(s) deleted for function or rule f
                                                                   Type: Void
(5) -> guess([f n for n in 0..20], maxLevel==2)
   Compiling function f with type NonNegativeInteger -> Fraction(
      Polynomial(Integer))

                   s   - 1
                    21      (4p   + 6)x
                    ++-++      20
                 2x  | |    -----------
                     | |              2
         n - 1     p  = 0   (p   + 2)y
          --+       20        20
   (5)  [ >      ---------------------- + 1]
          --+               2
         s  = 0            y
          21
                                              Type: List(Expression(Integer))
(6) -> guessPRec [f n for n in 0..20]

   (6)
   [
     [
       f(n):
                   2                      2
           (n + 2)y f(n + 2) + ((- n - 2)y  + (- 4n - 6)x)f(n + 1)
         +
           (4n + 6)x f(n)
           =
           0
       ,
                      2
                     y  + 2x
      f(0)= 1, f(1)= -------]
                         2
                        y
     ]
                                              Type: List(Expression(Integer))