Rationals r(n):= A123746(n)/A046161(n)
 
 r(n) = sum(((-1)^k)*binomial(2*k,k)/4^k,k=0..n) 

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

 r(n), n=0..30:
 
 [1, 1/2, 7/8, 9/16, 107/128, 151/256, 835/1024, 1241/2048, 26291/32768, 
 
  40427/65536, 207897/262144, 327615/524288, 3296959/4194304, 5293843/8388608,

  26189947/33554432, 42685049/67108864, 1666461763/2147483648, 

  2749521971/4294967296, 13266871709/17179869184, 22115585443/34359738368, 

  211386315749/274877906944, 355490397193/549755813888, 

  1684973959237/2199023255552, 2855358497999/4398046511104, 

  53747636888759/70368744177664, 91693947972799/140737488355328, 

  428765608509709/562949953421312, 735847502916411/1125899906842624, 

  6842866348426343/9007199254740992, 11806528540631371/18014398509481984, 

  54617650510329323/72057594037927936]


  Numerators are  A123746:

  [1, 1, 7, 9, 107, 151, 835, 1241, 26291, 40427, 207897, 327615, 3296959, 

  5293843, 26189947, 42685049, 1666461763, 2749521971, 13266871709, 

  22115585443, 211386315749, 355490397193, 1684973959237, 2855358497999, 

  53747636888759, 91693947972799, 428765608509709, 735847502916411, 

  6842866348426343, 11806528540631371, 54617650510329323]

  
  Denominators are A046161:
  
  [1, 2, 8, 16, 128, 256, 1024, 2048, 32768, 65536, 262144, 524288, 4194304, 

  8388608, 33554432, 67108864, 2147483648, 4294967296, 17179869184, 

  34359738368, 274877906944, 549755813888, 2199023255552, 4398046511104, 

  70368744177664, 140737488355328, 562949953421312, 1125899906842624, 

  9007199254740992, 18014398509481984, 72057594037927936]


  lim(r(n),n->infty) = s = 1/sqrt(2) from the expansion 1/sqrt(1-x) which

  has coefficients binomial(2*k)/4^k = A001790(k)/A046161(k) (central binomial

  coefficients scaled by powers of 4) valid for |x|<1 and also x=-1. due to 

  Abel's limit theorem and the convergence of the alternating series 

  s = lim(r(n),n->infty) shown with Leibniz' criterion. 

  The sequence b(k):=binomial(2*k)/4^k = (2*k-1)!!/(2*k)!! is monotonically 
 
  falling and has limit 0, shown with the help of the inequality 1+x <= exp(x)

  for all real x. (Cf. H. Heuser: Analysis I, pp 382-3).


  The values of the partial sums for n=10^k, k=0,1,...4, are:
  
  [.5000000000, .7930641174, .7352107634, .7160240574, .7099276233]     
 
  This should be compared with the limit

  1/sqrt(2) =   0.7071067810  (Maple 10digits). 
 
 
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