Rationals r(n) = A130547(n)/A130548(n), n>=1.

r(n):= 6*(sum(1/binomial(2*k,k),k=1..n) - 1/3).

r(n), n=1..30:
 
[1, 2, 23/10, 167/70, 253/105, 5581/2310, 13201/5460, 48413/20020, 823063/340340, 15638407/6466460, 1117033/461890, 89921239/37182145, 256917887/106234700, 60848977/25160850, 134111147453/55454513400, 4157445588203/1719089915400, 1385815197541/573029971800, 9700706385439/4011209802600, 358926136286437/148414762696200, 358926136292897/148414762696200, 474708760905697/196290492598200, 632786778288040231/261655226633400600, 949180167432346363/392482839950100900, 44611467869323711259/18446693477654742300, 156140137542636053869/64563427171791598050, 34697808342808185557/14347428260398132900, 3677967684337672355867/1520827395602202087400, 19668276386832478871/8132766821402150200, 1838983842168836926181/760413697801101043700, 217000093375922761841633/89728816340529923156600].


The numerators are A130547(n), n=1..30:

[1, 2, 23, 167, 253, 5581, 13201, 48413, 823063, 15638407, 1117033, 89921239, 256917887, 60848977, 134111147453, 4157445588203, 1385815197541, 9700706385439, 358926136286437, 358926136292897, 474708760905697, 632786778288040231, 949180167432346363, 44611467869323711259, 156140137542636053869, 34697808342808185557, 3677967684337672355867, 19668276386832478871, 1838983842168836926181, 217000093375922761841633].

The denominators are A130548(n), n=1..30: 

[1, 1, 10, 70, 105, 2310, 5460, 20020, 340340, 6466460, 461890, 37182145, 106234700, 25160850, 55454513400, 1719089915400, 573029971800, 4011209802600, 148414762696200, 148414762696200, 196290492598200, 261655226633400600, 392482839950100900, 18446693477654742300, 64563427171791598050, 14347428260398132900, 1520827395602202087400, 8132766821402150200, 760413697801101043700, 89728816340529923156600].

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The limit is 4*Pi*sqrt(3)/9

r(n) for n=10^k, k=0,1,2,3: (maple10, 15 digits):
[1., 2.41838764950220, 2.41839915231229, 2.41839915231229]


This should be compared with the value of the limit which is 
4*Pi*sqrt(3)/9 = 2.41839915231229 (maple10, 15 digits).  

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