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A084254
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Decimal expansion of Sum_{k>=1} 1/(k*(exp(2*Pi*k)-1)).
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9
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0, 0, 1, 8, 7, 2, 6, 8, 2, 4, 4, 9, 7, 6, 8, 5, 4, 6, 1, 1, 5, 6, 3, 8, 5, 7, 9, 4, 7, 9, 9, 6, 1, 3, 9, 8, 8, 6, 9, 1, 6, 2, 8, 9, 5, 6, 5, 2, 6, 1, 9, 5, 6, 3, 8, 4, 1, 3, 3, 1, 5, 7, 4, 5, 3, 7, 8, 8, 4, 3, 1, 9, 5, 1, 7, 0, 9, 8, 0, 2, 2, 6, 7, 5, 1, 7, 0, 7, 2, 7, 8, 4, 0, 2, 4, 5, 6, 7, 9, 7, 9, 9, 8, 7
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OFFSET
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0,4
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REFERENCES
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Bruce C. Berndt, Ramanujan Notebook part II, Infinite series, Springer Verlag, 1989, pp. 280-281.
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LINKS
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FORMULA
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Equals log(4/Pi)/4 - Pi/12 + log(Gamma(3/4)).
This is the case k=1, m=2 of the Plouffe sum S(k,m) = Sum_{n >= 1} 1/(n^k*(exp(m*Pi*n)-1)).
Pi = 72*S(1,1) - 96*S(1,2) + 24*S(1,4). (End)
Equals Sum_{k>=1} sigma(k)/(k*exp(2*Pi*k)). - Amiram Eldar, Jun 05 2023
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EXAMPLE
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0.00187268244976854611563857947996139886916289565261...
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MATHEMATICA
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digits = 104; S[1, 2] = NSum[1/(n*(Exp[2*Pi*n] - 1)), {n, 1, Infinity}, WorkingPrecision -> digits+10, NSumTerms -> digits]; RealDigits[S[1, 2], 10, digits] // First (* Jean-François Alcover, Mar 02 2015 *)
Join[{0, 0}, RealDigits[Log[4/Pi]/4 - Pi/12 + Log[Gamma[3/4]], 10, 100][[1]]] (* Amiram Eldar, May 21 2022 *)
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PROG
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(PARI) 1/4*log(4/Pi)-Pi/12+log(gamma(3/4))
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CROSSREFS
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Cf. A255695 (S(1,1)), A255697 (S(1,4)), A255698 (S(3,1)), A255699 (S(3,2)), A255700 (S(3,4)), A255701 (S(5,1)), A255702 (S(5,2)), A255703 (S(5,4)).
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KEYWORD
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AUTHOR
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STATUS
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approved
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