Rationals  r(n):= A123747(n)/A123748(n)

 r(n)= sum(binomial(2*k,k)/5^k,k=0..n) in lowest terms.

 r(n)=sum(((2*k-1)!!/(2*k)!!)*(4/5)^k),k=0..n),n>=0, 
 with the double factorials A001147 and A000165.


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

 [1, 7/5, 41/25, 9/5, 239/125, 6227/3125, 32059/15625, 163727/78125, 

  166301/78125, 841229/390625, 21215481/9765625, 106782837/48828125, 

  536618341/244140625, 538698461/244140625, 172897/78125, 

  13538601629/6103515625, 67813224223/30517578125, 339532842359/152587890625, 

  339895847771/152587890625, 1700893049407/762939453125, 

  42549895540939/19073486328125, 212857129279583/95367431640625, 

  1064706466190659/476837158203125, 1065035803419763/476837158203125, 

  5326468921246139/2384185791015625, 665935025762205127/298023223876953125, 

  3330171047343973739/1490116119384765625, 

  16652802176145516807/7450580596923828125, 

  3330866382853133779/1490116119384765625, 

  83277673024628252683/37252902984619140625, 

  2082060090197271178499/931322574615478515625]


 The numerators are A123747(n), n=0..30:

 [1, 7, 41, 9, 239, 6227, 32059, 163727, 166301, 841229, 21215481, 106782837, 

  536618341, 538698461, 172897, 13538601629, 67813224223, 339532842359, 

  339895847771, 1700893049407, 42549895540939, 212857129279583, 

  1064706466190659, 1065035803419763, 5326468921246139, 665935025762205127, 

  330171047343973739, 16652802176145516807, 3330866382853133779, 

  83277673024628252683, 2082060090197271178499]


 The denominators are A123748(n), n=0..30:

 [1, 5, 25, 5, 125, 3125, 15625, 78125, 78125, 390625, 9765625, 48828125, 

  244140625, 244140625, 78125, 6103515625, 30517578125, 152587890625, 

  152587890625, 762939453125, 19073486328125, 95367431640625, 
 
  476837158203125, 476837158203125, 2384185791015625, 298023223876953125, 

  1490116119384765625, 7450580596923828125, 1490116119384765625, 

  37252902984619140625, 931322574615478515625]  

  
  The series s:=lim(r(n),n->infinity) has value sqrt(5). This is the convergent

  binomial expansion (1-x)^(-1/2) for x=4/5.

  The values for r(n) for n=10^k, k=0,1,...,3, are (10 digits Maple10) 

  [1.400000000, 2.172465254, 2.236067977, 2.236067977].

  This should be compared with the value (10 digits Maple10)
  
  sqrt(5) = 2*phi-1 =  2.236067977.


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