r(n)=  A120994(n)/ A120995(n) 

 Rationals r(n):=(3/4)*W(n), with W(n):= (((2*n)!!/(2*n-1)!!)^2)/(2*n+1) = (((4^n)/binomial(2*n,n))^2)/(2*n+1).

 limit(n to infinity,  W(n)) = pi/2 (John Wallis' product formula from his 1656 book 'Arithmetica infinitorum').

 Hence, limit(n to infinity, r(n)) = 3*pi/8 = 1.178097245...


 r(n), n=1..30:

 [1, 16/15, 192/175, 4096/3675, 16384/14553, 262144/231231, 1048576/920205, 268435456/234652275, 
3221225472/2807136475, 17179869184/14933966047, 68719476736/59612442981, 13194139533312/11425718238025, 
17592186044416/15211755050625, 281474976710656/243077636829375, 1125899906842624/971230202264925, 
1152921504606846976/993568496917018275, 4611686018427387904/3970836034391543625, 
73786976294838206464/63484353883173444375, 295147905179352825856/253761558596729308125, 
18889465931478580854784/16230589287846806547675, 75557863725914323419136/64885553094045169940025, 
1208925819614629174706176/1037632605264276064743375, 4835703278458516698824704/4148568922748476137868125, 
1237940039285380274899124224/1061572692121082282834476875, 
4951760157141521099596496896/4244592252176935399685372337, 
79228162514264337593543950336/67888360104344712339346517319, 
950737950171172051122527404032/814380945284628956663354312695, 
20282409603651670423947251286016/17367920159641576728840923607475, 
81129638414606681695789005144064/69451029128269467941845453141425, 
3894222643901120721397872246915072/3332723384435224201636023811413181]
 
 
 Some numerical values of r(n) are, for n= 10^k, k=0..4: (maple10, 10 digits)

 [1., 1.150388927, 1.175170309, 1.177802905, 1.178067795]

 to be compared with  3*pi/8 = 1.178097245...

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  The Wallis sequence W(n) with limit pi/2 = 1.570796327 is, for n=1..30:

  [4/3, 64/45, 256/175, 16384/11025, 65536/43659, 1048576/693693, 4194304/2760615, 1073741824/703956825, 
4294967296/2807136475, 68719476736/44801898141, 274877906944/178837328943, 17592186044416/11425718238025, 
70368744177664/45635265151875, 1125899906842624/729232910488125, 4503599627370496/2913690606794775, 
4611686018427387904/2980705490751054825, 18446744073709551616/11912508103174630875, 
295147905179352825856/190453061649520333125, 1180591620717411303424/761284675790187924375, 
75557863725914323419136/48691767863540419643025, 302231454903657293676544/194656659282135509820075, 
4835703278458516698824704/3112897815792828194230125, 19342813113834066795298816/12445706768245428413604375, 
4951760157141521099596496896/3184718076363246848503430625, 
19807040628566084398385987584/12733776756530806199056117011, 
316912650057057350374175801344/203665080313034137018039551957, 
1267650600228229401496703205376/814380945284628956663354312695, 
81129638414606681695789005144064/52103760478924730186522770822425, 
324518553658426726783156020576256/208353087384808403825536359424275, 
5192296858534827628530496329220096/3332723384435224201636023811413181]

Some numerical values for W(n) are, for n=10^k, k=0..4: (maple10, 10 digits)

[1.333333333, 1.533851903, 1.566893745, 1.570403873, 1.570757059]

to be compared with pi/2 =  1.570796327...

##########################################################################################################

 Wallis' product formula for pi/2 (or 3*pi/8) leads to the asymptotic formula for C(k)/4^k with the Catalan 
 
 numbers C(n) = binomial(2*n,n)/(n+1) = A000108(n): take the square of the reciprocal formula, with (2*n+1) 
 
 replaced by 2*n (for large n):

 C(n)/4^n is asympotically equal to  1/(sqrt(pi)*n^(3/2)) .

 See also Exercise 9.8 in the Graham-Knuth-Patashnik book.
 
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