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 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)). 2
 1, 3, 299, 1691, 4451729, 13446833, 16372396819, 208298035171, 1669160962863, 446401251163753, 6516008708737202119, 44233149340111747277, 5029067414956952883994601, 5810809342741928035310687, 46442062699559407155897191, 1018306138326248284055588777, 369103117042133718901423551221401 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 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 MAPLE with(numtheory): P:=proc(q); numer(add(add((k+1)/(m-k+1)^2*((binomial(2*k, k)/(k+1))/(2^(2*k)))^2, k=0..m)*add((k+1)/(q-m-k+1)^2*((binomial(2*k, k)/(k+1))/(2^(2*k)))^2, k=0..q-m), m=0..q)); end: seq(P(i), i=0..100); # Paolo P. Lava, Feb 14 2017 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] PROG (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 CROSSREFS Cf. A281070, A280723, A282196 (denominators). Cf. A000108 (Catalan), A019693 (2 Pi/3). Sequence in context: A212798 A157579 A104821 * A303388 A119065 A119069 Adjacent sequences:  A282192 A282193 A282194 * A282196 A282197 A282198 KEYWORD nonn,frac AUTHOR Ralf Steiner, Feb 08 2017 STATUS approved

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Last modified January 19 17:45 EST 2019. Contains 319309 sequences. (Running on oeis4.)