This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A385612 #26 Jul 05 2025 09:59:11 %S A385612 2,3,9,9,7,1,0,3,1,8,8,0,1,3,7,0,7,9,5,8,9,8,7,2,1,9,5,2,7,7,4,1,0,0, %T A385612 5,6,6,1,8,9,1,1,3,9,9,3,4,9,2,1,7,0,3,4,2,4,9,7,6,0,0,9,3,3,3,0,4,6, %U A385612 3,8,2,9,3,8,6,3,3,4,4,9,9,1,3,8,2,8,6,1,8,2,2,7,5,7,8,1,3,3,4,6,9,4,9,0,3 %N A385612 Decimal expansion zeta''''(0) (negated). %C A385612 n-th derivative of zeta function at 0 is close to -n!, which here is the present constant close to 4! = 24. %H A385612 Tom M. Apostol, <a href="https://doi.org/10.1090/S0025-5718-1985-0771044-5">Formulas for higher derivatives of the Riemann zeta function</a>, Mathematics of Computation 44 (1985), pp. 223-232. %F A385612 Equals -3*gamma^4/2 - gamma^2*Pi^2/4 + 19*Pi^4/480 - 4*gamma^3*log(2*Pi) -3*gamma^2*log(2*Pi)^2 + Pi^2*log(2*Pi)^2/4 + log(2*Pi)^4/2 - 6*gamma^2*StieltjesGamma(1) - Pi^2*StieltjesGamma(1)/2 - 12*gamma*log(2*Pi)* StieltjesGamma(1) - 6*log(2*Pi)^2*StieltjesGamma(1) - 6*gamma*StieltjesGamma(2) - 6*log(2*Pi)*StieltjesGamma(2) - 2*StieltjesGamma(3) + 4*log(2*Pi)*zeta(3). %e A385612 23.997103188013707958987219527741... %p A385612 evalf(-Zeta(4, 0), 120); # _Vaclav Kotesovec_, Jul 04 2025 %t A385612 RealDigits[-3 EulerGamma^4/2 - EulerGamma^2 Pi^2/4 + 19 Pi^4/480 - 4 EulerGamma^3 Log[2 Pi] - 3 EulerGamma^2 Log[2Pi]^2 + Pi^2 Log[2 Pi]^2/4 + Log[2 Pi]^4/2 - 6 EulerGamma^2 StieltjesGamma[1] - Pi^2 StieltjesGamma[1]/2 - 12 EulerGamma Log[2 Pi] StieltjesGamma[1] - 6 Log[2 Pi]^2 StieltjesGamma[1] - 6 EulerGamma StieltjesGamma[2] - 6 Log[2Pi] StieltjesGamma[2] - 2 StieltjesGamma[3] + 4 Log[2 Pi] Zeta[3],10,105][[1]] %o A385612 (PARI) -zeta''''(0) %Y A385612 Cf. A075700, A257549, A261508. %Y A385612 Cf. A001620, A061444, A082633, A086279, A086280. %K A385612 nonn,cons %O A385612 2,1 %A A385612 _Artur Jasinski_, Jul 04 2025