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a(n) is the numerator of 6 * Sum_{k=0..n} ((k+1)/(n-k+1)^2) * (Catalan(k)/(2^(2*k+1)))^2.
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%I #35 Jan 15 2017 13:22:07

%S 3,9,109,1037,91027,1540981,447810157,147053171,503445581741,

%T 16337573574319,88973047698967,3588920671411951,2314594755016141847,

%U 20685050199210758743,2160689714871889935101,121435710295138581181033,16427863327419202412927713

%N a(n) is the numerator of 6 * Sum_{k=0..n} ((k+1)/(n-k+1)^2) * (Catalan(k)/(2^(2*k+1)))^2.

%C The series a(n)/A280723(n) is absolutely convergent to Pi.

%t a[n_]=6(Sum[(1/(n-k+1)^2)((CatalanNumber[k])/(2^(2k+1)))^2(k+1),{k,0, n}]); Numerator /@a/@ Range[0,10]

%Y Cf. A000108 (Catalan), A280723 (denominators).

%K nonn,frac

%O 0,1

%A _Ralf Steiner_, Jan 14 2017