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A268046 Decimal expansion of Sum_{k>0} (k+1)/(k*((k+1)^2+1)).

%I A268046 #24 Mar 06 2025 04:43:57
%S A268046 8,7,4,2,7,0,0,1,6,4,9,6,2,9,5,5,0,6,0,0,6,2,2,0,6,8,3,7,7,7,1,5,6,8,
%T A268046 2,1,9,8,2,0,3,9,2,8,7,0,5,3,6,4,0,9,4,3,8,0,6,9,0,9,7,2,1,6,9,6,9,6,
%U A268046 3,5,4,4,7,2,7,6,2,7,6,7,3,0,3,1,9,9,1,6,6,2,2,7,3,2,9,7,9,5,6,6
%N A268046 Decimal expansion of Sum_{k>0} (k+1)/(k*((k+1)^2+1)).
%C A268046 Also, decimal expansion of Integral_{x=0..1} (2 - (1+i)*x^(1-i) - (1-i)*x^(1+i))/(4 - 4*x) dx, where i is the imaginary unit.
%F A268046 Equals (1 + i)*(H(1-i) - i*H(1+i))/4, where H(z) is a harmonic number with complex argument.
%F A268046 Equals (Psi(i-1)-Psi(1)-i-1)/2 - Pi*(i-1)*coth(Pi)/4, where Psi(x) is the digamma function. - _Peter Luschny_, Jan 27 2016
%e A268046 .8742700164962955060062206837771568219820392870536409438069097216969635...
%p A268046 ((1+I)*(harmonic(1-I)-I*harmonic(1+I)))/4:
%p A268046 Re(evalf(%, 101)); # _Peter Luschny_, Jan 27 2016
%t A268046 (1 + I)*(HarmonicNumber[1 - I] - I*HarmonicNumber[1 + I])/4 // Re // RealDigits[#, 10, 100]& // First (* _Jean-François Alcover_, Jan 26 2016 *)
%o A268046 (Sage)
%o A268046 # Warning: Floating point calculation. Adjust precision as needed
%o A268046 # and use some guard digits!
%o A268046 from mpmath import mp, chop, psi, pi, coth
%o A268046 mp.dps = 109; mp.pretty = True
%o A268046 chop((psi(0,I-1)-psi(0,1)-I-1)/2-pi*(I-1)*coth(pi)/4) # _Peter Luschny_, Jan 27 2016
%Y A268046 Cf. A268086: (1-i)*(H(1-i)+i*H(1+i))/4.
%K A268046 nonn,cons
%O A268046 0,1
%A A268046 _Bruno Berselli_, Jan 26 2016