Rationals r(n):=A119951(n)/A120069(n) = sum(C(k)/4^(k-1),k=1..n), n>=1..30:

[1, 3/2, 29/16, 65/32, 281/128, 595/256, 9949/4096, 20613/8192, 84883/32768, 173965/65536, 1421113/524288, 2894229/1048576, 11762641/4194304, 23859587/8388608, 773201629/268435456, 1564082093/536870912, 6321150767/2147483648, 12761711209/4294967296, 102977321267/34359738368, 207595672639/68719476736, 836499257311/274877906944, 1684433835077/549755813888, 27122471168057/8796093022208, 54567418372945/17592186044416, 219485160092143/70368744177664, 441266239318305/140737488355328, 3547513302275441/1125899906842624, 7127995098519677/2251799813685248, 28637260671160129/9007199254740992, 57512957998701027/18014398509481984]

The numerator sequence is A119951(n), for n=1..30:

[1, 3, 29, 65, 281, 595, 9949, 20613, 84883, 173965, 1421113, 2894229, 11762641, 23859587, 773201629, 1564082093, 6321150767, 12761711209, 102977321267, 207595672639, 836499257311, 1684433835077, 27122471168057, 54567418372945, 219485160092143, 441266239318305, 3547513302275441, 7127995098519677, 28637260671160129, 57512957998701027]


The denominator sequence is A120069(n), for n=1..30:

[1, 2, 16, 32, 128, 256, 4096, 8192, 32768, 65536, 524288, 1048576, 4194304, 8388608, 268435456, 536870912, 2147483648, 4294967296, 34359738368, 68719476736, 274877906944, 549755813888, 8796093022208, 17592186044416, 70368744177664, 140737488355328, 1125899906842624, 2251799813685248, 9007199254740992, 18014398509481984]

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The values (maple10, 10 digits) for the rationals r(n) are, for n=10^k, k=1,2,..,5 are:


[2.654495239, 3.551443791, 3.857359196, 3.954867654, 3.985727096]]


From the expansion of sqrt(1+x) = 1 + (x/2)*sum((C_k)*(-x/4)^k,k=0..infty), valid for |x|<=1, one finds for x=-1:

s:=sum(C(k)/2^(2*(k-1)),k=1..infty) with the value s=4.


The value of the sum over the asymptotic values for C(k)/4^{k-1}, i.e.  4/(sqrt(Pi)*k^(3/2)), 

for k=1..infinity, is 4*Zeta(3/2)/sqrt(Pi) which is 5.895499840 (maple10, 10 digits).

The (C_k)/4^k asymptotics follows from the taking the square-root of the reciprocal of Wallis' product formula (1656) 

for Pi/2.


The numerator and denominator numbers become large. For example, A119951(10^4) has 6204 digits,

and A119951(10^5) has 60209 digits.


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