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Whilst trying to solve a combinatorics problem I am faced with summing this series:

1+ 2C_1 2/(3^2) + 4C_2 (2^2)/(3^4) + 6C_3 (2^3)/(3^6)+ ... + 2nC_n (2^n)/(3^(2n))+...

Where 4C_2 is 4 choose 2.

Any idea how to approach this problem?

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It's a hypergeometric series; and your sum has a nice closed-form. Try it on Wolfram Alpha. –  J. M. Aug 31 '10 at 11:56
    
In particular, the series can be turned into \sum_{j=0}^\infty \binom{-1/2}{j}(-4z)^j where z=2/9. –  J. M. Aug 31 '10 at 12:03
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2 Answers

up vote 4 down vote accepted

The generating function of the central binomial coefficients is $$\sum_{n=0}^{\infty}\binom{2n}{n}x^n=\frac{1}{\sqrt{1-4x}}$$ and so the value of your series is 3.

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Wow, quite close. :) –  J. M. Aug 31 '10 at 12:04
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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.


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))
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