A282195 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).

1, 3, 299, 1691, 4451729, 13446833, 16372396819, 208298035171, 1669160962863, 446401251163753, 6516008708737202119, 44233149340111747277, 5029067414956952883994601, 5810809342741928035310687, 46442062699559407155897191, 1018306138326248284055588777, 369103117042133718901423551221401

Offset: 0

Author

Ralf Steiner, Feb 08 2017

Keywords

• nonn
• frac

Comments

The series a(n)/A282196(n) is absolutely convergent to (2/3 Pi)^2.

Links

• Paolo P. Lava, Table of n, a(n) for n = 0..100

Crossrefs

Cf. A281070, A280723, A282196 (denominators).

Cf. A000108 (Catalan), A019693 (2 Pi/3).

Programs

Mathematica

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]

PARI

C(n) = binomial(2*n,n)/(n+1);
b(n) = sum(k=0, n, ((k+1)/(n-k+1)^2) * (C(k)/(2^(2*k)))^2);
a(n) = numerator(sum(k=0, n, b(k)*b(n-k))); \\ Michel Marcus, Feb 11 2017