A261506 -id:A261506 - cp's OEIS frontend

A349252 Decimal expansion of Sum_{k>=1} (-1)^k * log(k) / k^4.

0, 3, 3, 4, 7, 8, 8, 0, 4, 5, 7, 8, 5, 6, 5, 0, 6, 6, 3, 8, 5, 9, 5, 6, 8, 5, 4, 7, 8, 8, 7, 3, 7, 7, 9, 9, 7, 1, 3, 7, 5, 9, 7, 3, 0, 4, 0, 5, 7, 3, 4, 9, 7, 4, 8, 2, 8, 6, 6, 5, 7, 6, 4, 2, 8, 8, 6, 8, 3, 6, 2, 2, 5, 2, 7, 9, 5, 8, 8, 3, 8, 1, 0, 7, 9, 5, 3, 4, 7, 4, 7, 5, 8, 6, 5, 8, 6, 4, 8, 6, 2, 2, 8, 2, 6, 6, 5, 1, 1, 1, 1, 2, 1, 8, 5, 5, 1, 7, 9, 8, 3

Offset: 0

Ilya Gutkovskiy, Nov 12 2021

First derivative of the Dirichlet eta function at 4.

			0.0334788045785650663859568547887377997137597304057...

Eric Weisstein's World of Mathematics, Dirichlet Eta Function

Cf. A013662, A091812, A210593, A256358, A261506, A267315, A349220.

Flatten[{0, RealDigits[(Pi^4 Log[2] + 630 Zeta'[4])/720, 10, 120][[1]]}]

sumalt(k=1, (-1)^k * log(k) / k^4) \\ Michel Marcus, Nov 12 2021

Equals (Pi^4 * log(2) + 630 * zeta'(4)) / 720.

A324021 Decimal expansion of integral_{x>0} tanh(x)^3/x^2 dx.

Original entry on oeis.org

1, 1, 5, 4, 7, 8, 5, 3, 1, 3, 3, 2, 3, 1, 7, 6, 2, 6, 4, 0, 5, 9, 0, 7, 0, 4, 5, 1, 9, 4, 1, 5, 2, 6, 1, 4, 7, 5, 3, 5, 2, 3, 7, 0, 9, 2, 4, 5, 0, 8, 9, 2, 4, 8, 9, 0, 9, 9, 1, 1, 0, 2, 2, 0, 2, 9, 1, 1, 3, 7, 8, 5, 7, 0, 5, 6, 1, 1, 9, 1, 3, 1, 9, 5, 4, 6, 5, 8, 5, 1, 8, 9, 5, 8, 6, 4, 4, 7, 5, 7, 7

Offset: 1

Jean-François Alcover, Sep 01 2019

			1.15478531332317626405907045194152614753523709245089248909911022029...

Igor Khavkine, Is there a closed form for integral_{x>0} tanh(x)^3/x^2 dx, MathOverflow, 2017.
Jing Li and Wenchang Chu, Integrals of Hyperbolic Tangent Function, arXiv:2310.19847 [math.GM], 2023. Has similar integrals.

Cf. A074962 (Glaisher constant), A261506 (-zeta'(4)).

RealDigits[N[5/6 - EulerGamma - 19 Log[2]/15 + 12 Log[Glaisher] - Log[Pi] + 90 Zeta'[4]/Pi^4, 101]][[1]]

5/6 - gamma - 19*log(2)/15 + 12*log(A) - log(Pi) + 90*zeta'(4)/Pi^4, where gamma is the Euler-Mascheroni constant, and A is the Glaisher constant.

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A349252 Decimal expansion of Sum_{k>=1} (-1)^k * log(k) / k^4.

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