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A268086 Decimal expansion of Sum_{k>0} 1/(k*((k+1)^2+1)).

%I A268086 #25 Mar 06 2025 04:43:47
%S A268086 2,9,7,5,9,5,9,6,9,0,2,7,7,1,4,3,3,1,8,7,2,1,6,9,8,8,9,0,2,7,1,5,6,3,
%T A268086 3,1,5,3,6,3,8,3,0,2,0,6,4,9,8,2,4,2,7,8,2,3,1,8,4,7,2,3,7,3,0,6,8,0,
%U A268086 9,2,9,6,8,0,9,3,1,7,6,5,1,2,8,8,4,2,6,1,1,0,5,1,3,9,0,2,4,6,4,7
%N A268086 Decimal expansion of Sum_{k>0} 1/(k*((k+1)^2+1)).
%C A268086 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 A268086 Equals (1 - i)*(H(1-i) + i*H(1+i))/4, where H(z) is a harmonic number with complex argument.
%F A268086 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 A268086 .297595969027714331872169889027156331536383020649824278231847237306809...
%p A268086 ((1-I)*(harmonic(1-I) + I*harmonic(1+I)))/4:
%p A268086 Re(evalf(%, 106)); # _Peter Luschny_, Jan 27 2016
%t A268086 (1 - I)*(HarmonicNumber[1 - I] + I*HarmonicNumber[1 + I])/4 // Re // RealDigits[#, 10, 100]& // First (* _Jean-François Alcover_, Jan 26 2016 *)
%o A268086 (Sage)
%o A268086 # Warning: Floating point calculation. Adjust precision as needed
%o A268086 # and use some guard digits!
%o A268086 from mpmath import mp, chop, psi, coth, pi
%o A268086 mp.dps = 108; mp.pretty = True
%o A268086 chop((psi(0,I-1)-psi(0,1)-I+1)/2-pi*(I+1)*coth(pi)/4) # _Peter Luschny_, Jan 27 2016
%Y A268086 Cf. A062158: numbers of the form k*((k+1)^2+1), with k>-2.
%Y A268086 Cf. A268046: (1+i)*(H(1-i)-i*H(1+i))/4.
%K A268086 nonn,cons
%O A268086 0,1
%A A268086 _Bruno Berselli_, Jan 26 2016