W. Lang, Oct 13, 2008

A145557 numerators of the rationals r(n):=sum(((-1)^(k+1))/(k*binomial(2*k,k)),k=1..n), n>=1, 
(in lowest terms) for  n=1..25:

[1, 5, 13, 361, 31, 1193, 31021, 34467, 5273479, 1821745, 220211, 230450795, 2880634987, 
1502939987, 5896829249, 12430516053889, 1381168450513, 3271188435379, 2299645470079393, 
459929094015491, 819873602375609, 810854992749436603, 311867304903633289, 31758487216019894591, 
24098581085326767].


A145558 1/2 of denominators of the rationals r(n):=sum(((-1)^(k+1))/binomial(2*k,k),k=1..n), n>=1, 
(in lowest terms) for  n=1..25:

[1, 6, 15, 420, 36, 1386, 36036, 40040, 6126120, 2116296, 255816, 267711444, 3346393050, 
1745944200, 6850263420, 14440355289360, 1604483921040, 3800093497200, 2671465728531600, 
534293145706320, 952435607563440, 941958815880242160, 362291852261631600, 36893386955309484600, 
27994981971508552].

The denominators of r(n) are 2*A145558, for n=1..25:

[2, 12, 30, 840, 72, 2772, 72072, 80080, 12252240, 4232592, 511632, 535422888, 6692786100, 
3491888400, 13700526840, 28880710578720, 3208967842080, 7600186994400, 5342931457063200, 
1068586291412640, 1904871215126880, 1883917631760484320, 724583704523263200, 73786773910618969200, 55989963943017104].



Rationals r(n)= A145557(n)/(2*A145558(n)), n=1..25: 
[1/2, 5/12, 13/30, 361/840, 31/72, 1193/2772, 31021/72072, 34467/80080, 5273479/12252240, 
1821745/4232592, 220211/511632, 230450795/535422888, 2880634987/6692786100, 
1502939987/3491888400, 5896829249/13700526840, 12430516053889/28880710578720, 
1381168450513/3208967842080, 3271188435379/7600186994400, 2299645470079393/5342931457063200, 
459929094015491/1068586291412640, 819873602375609/1904871215126880, 
810854992749436603/1883917631760484320, 311867304903633289/724583704523263200, 
31758487216019894591/73786773910618969200, 24098581085326767/55989963943017104].


The values r(10^k), for k=0,..3 are (maple11, 10 digits) [.5000000000, .4304088369, 
.4304089410, .4304089410].

This should be compared with the limit n -> infinity which is 
2*(2*phi-1)*ln(phi)/5) with the golden section phi:=(1+sqrt(5))/2 which 
is approximatly (maple11, 10 digits) 0.4304089412 .


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