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%I #7 Jun 10 2015 05:20:35
%S 1,1,2,1,2,2,4,223,854,4762,1444,2324506,35548,140343676,21047728264,
%T 88824427,160465442,96633020222386,18457536052,1397584483920886,
%U 885721299987868,2758893844640044,3793843972393624,56271391915038457502,480348904858674456484,85734822774933179463764,2140418040120050844508958552
%N Numerators of a rational sequence related to the closed form evaluation of a Gaussian integer zeta sum.
%H Jonathan Borwein, <a href="http://www.siam.org/journals/problems/downloadfiles/05-002s.pdf">Also Eisenstein</a>
%F p(1) = 1/5, p(n) = (3/((4n+1)*(2n-3)))*Sum_{k=1..n-1} p(k)*p(n-k).
%F The closed form of the Gaussian integer zeta sum in question is zeta_G(4n) = p(n)*L^(4n)/(4n-1), where L is the lemniscate constant.
%e Sequence of fractions begins:
%e 1/5, 1/75, 2/4875, 1/82875, 2/6215625, 2/242409375, 4/19527421875, 223/ 44815433203125, ...
%t p[1] = 1/5; p[n_] := p[n] = (3/((4n+1)*(2n-3)))*Sum[p[k]*p[n-k], {k, 1, n-1}]; Table[p[n] // Numerator, {n, 1, 30}]
%Y Cf. A062539 (lemniscate constant), A258784 (denominators).
%K nonn,frac,easy
%O 1,3
%A _Jean-François Alcover_, Jun 10 2015