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%I #37 Aug 20 2022 13:27:34
%S 1,-1,1,1,1,1,1,1,-1,-1,-1,-1,-1,-1,-1,-1,-1,1,1,1,1,1,1,1,1,1,1,1,1,
%T 1,1,1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,
%U 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1
%N a(n) is the sign of Maslanka's constant A(n).
%C In 1997 Krzysztof Maslanka published a rapidly converging series giving values of the zeta function which uses coefficients A(k), involving even powers of Pi.
%C The sum of all positive A(n) equals B=2.143505674422168409234504232295468835062643090089323...
%C The sum of all negative A(n) equals C=-1.643505674422168409234504232295468835062643090089323...
%C B+C = 1/2.
%D K. Maslanka, The Beauty of Nothingness: Essay on the Zeta Function of Riemann, Acta Cosmologica XXIII, pp. 13-18 (1998).
%H Krzysztof Maslanka, <a href="/A355437/b355437.txt">Table of n, a(n) for n = 0..60000</a>
%H K. Maslanka, <a href="https://arxiv.org/abs/math-ph/0105007">Hypergeometric-like Representation of Zeta function of Riemann</a>, arXiv:math-ph/0105007, pp. 1-6 (1997).
%H K. Maslanka, <a href="https://arxiv.org/abs/math/0603713">Báez Duarte's Criterion for the Riemann Hypothesis and Rice's Integrals</a>, arXiv:math/0603713 [math.NT], pp. 1-16 (1997).
%H K. Maslanka and A. Kolezynski, <a href="http://dx.doi.org/10.12921/cmst.2022.0000014">The High Precision Numerical Calculation of Stieltjes Constants. Simple and Fast Algorithm</a>, Computational Methods in Science and Technology, Volume 28 (2) 2022, 47-59.
%F a(n) = sign(Sum_{j=0..n} (-1)^j*binomial(n, j)*(2*j + 1)*zeta(2*j + 2)).
%e A(0) = +1.644934067 so a(0)=+1,
%e A(1) = -1.602035634 so a(1)=-1,
%e A(2) = +0.2377099745 so a(2)=+1 etc.
%t A[k_] := Sum[(-1)^j Binomial[k, j] (2 j + 1) Zeta[2 j + 2], {j, 0, k}]; Table[Sign[N[A[n], 100]], {n, 0, 115}]
%Y Cf. A114523, A114524, A354835.
%K sign
%O 0
%A _Artur Jasinski_, Jul 02 2022
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