Rationals r(n)=A123749(n)/A124396(n)

 r(n):=sum(binomial(2*k,k)/9^k,k=0..n) in lowest terms.
 
 r(n)=sum((2*k-1)!!/((2*k)!!)*(4/9)^k),k=0..n),n>=0, 
 with the double factorials A001147 and A000165.

 r(n), n=0..30:

 [1, 11/9, 35/27, 965/729, 8755/6561, 8783/6561, 237449/177147, 2138185/1594323, 

  6415985/4782969, 519743405/387420489, 4677875401/3486784401, 14033861347/10460353203, 

  378916960525/282429536481, 3410263045325/2541865828329, 3410267502725/2541865828329, 

  30692424759805/22876792454961, 276231889624955/205891132094649, 

  828695755304725/617673396283947, 67124359204727825/50031545098999707, 

  604119244624305025/450283905890997363, 1812357749189196055/1350851717672992089, 

  48933659407527584965/36472996377170786403, 440402935369114585925/328256967394537077627, 

  146800978558018821575/109418989131512359209, 11890879266782591623475/8862938119652501095929,

  107017913415088947548803/79766443076872509863361, 

  321053740263634195718561/239299329230617529590083, 

  78016058886010048985258435/58149737003040059690390169, 

  702144529981739131468086355/523347633027360537213511521, 

  2106433589955239816570772745/1570042899082081611640534563, 

  56873706928909739628975725539/42391158275216203514294433201]  


  Numerators are A123749; for n=0..30:

  [1, 11, 35, 965, 8755, 8783, 237449, 2138185, 6415985, 519743405, 4677875401, 

  14033861347, 378916960525, 3410263045325, 3410267502725, 30692424759805, 276231889624955, 

  828695755304725, 67124359204727825, 604119244624305025, 1812357749189196055, 

  48933659407527584965, 440402935369114585925, 146800978558018821575, 

  11890879266782591623475, 107017913415088947548803, 321053740263634195718561, 

  78016058886010048985258435, 702144529981739131468086355, 2106433589955239816570772745, 

  56873706928909739628975725539]

   
  Denominators are A124396; for n=0..30:

  [1, 9, 27, 729, 6561, 6561, 177147, 1594323, 4782969, 387420489, 3486784401, 10460353203, 

   282429536481, 2541865828329, 2541865828329, 22876792454961, 205891132094649, 
  
   617673396283947, 50031545098999707, 450283905890997363, 1350851717672992089, 

   36472996377170786403, 328256967394537077627, 109418989131512359209, 

   8862938119652501095929, 79766443076872509863361, 239299329230617529590083, 

   58149737003040059690390169, 523347633027360537213511521, 1570042899082081611640534563, 

   42391158275216203514294433201]

   
   The values of the rational r(10^k), k=0..3 are (10 digits Maple10)

   [1.222222222, 1.341601563, 1.341640786, 1.341640786]  

   They should be compared with 

   3/sqrt(5) = (3/5)*sqrt(5) = 1.341640786 (Maple 10 digits).


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