A385612 - OEIS

A385612

Decimal expansion zeta''''(0) (negated).

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, 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, 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

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OFFSET

2,1

COMMENTS

n-th derivative of zeta function at 0 is close to -n!, which here is the present constant close to 4! = 24.

LINKS

Table of n, a(n) for n=2..106.

Tom M. Apostol, Formulas for higher derivatives of the Riemann zeta function, Mathematics of Computation 44 (1985), pp. 223-232.

FORMULA

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).

EXAMPLE

23.997103188013707958987219527741...

MAPLE

evalf(-Zeta(4, 0), 120); # Vaclav Kotesovec, Jul 04 2025

MATHEMATICA

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]]

PROG

(PARI) -zeta''''(0)

CROSSREFS

Cf. A075700, A257549, A261508.

Cf. A001620, A061444, A082633, A086279, A086280.

Sequence in context: A122665 A133066 A337998 * A131988 A092593 A368680

Adjacent sequences: A385609 A385610 A385611 * A385613 A385614 A385615

KEYWORD

nonn,cons

AUTHOR

Artur Jasinski, Jul 04 2025

STATUS

approved