|
%I #30 Mar 11 2024 01:49:37
%S 1,3,299,1691,4451729,13446833,16372396819,208298035171,1669160962863,
%T 446401251163753,6516008708737202119,44233149340111747277,
%U 5029067414956952883994601,5810809342741928035310687,46442062699559407155897191,1018306138326248284055588777,369103117042133718901423551221401
%N a(n) is the numerator of Sum_{m=0..n}(Sum_{k=0..m} ((k+1)/(m-k+1)^2) * (Catalan(k)/(2^(2*k)))^2)*(Sum_{k=0..n-m} ((k+1)/(n-m-k+1)^2) * (Catalan(k)/(2^(2*k)))^2).
%C The series a(n)/A282196(n) is absolutely convergent to (2/3 Pi)^2.
%H Paolo P. Lava, <a href="/A282195/b282195.txt">Table of n, a(n) for n = 0..100</a>
%t b[n_]=(Sum[((k+1)/(n-k+1)^2)((CatalanNumber[k])/(2^(2k)))^2, {k, 0, n}]); a[n_] = Sum[(b[k]*b[n - k]), {k, 0, n}]; Numerator /@a/@ Range[0, 10]
%o (PARI) C(n) = binomial(2*n,n)/(n+1);
%o b(n) = sum(k=0, n, ((k+1)/(n-k+1)^2) * (C(k)/(2^(2*k)))^2);
%o a(n) = numerator(sum(k=0, n, b(k)*b(n-k))); \\ _Michel Marcus_, Feb 11 2017
%Y Cf. A281070, A280723, A282196 (denominators).
%Y Cf. A000108 (Catalan), A019693 (2 Pi/3).
%K nonn,frac
%O 0,2
%A _Ralf Steiner_, Feb 08 2017
|
