A324996 - OEIS

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A324996

Decimal expansion of zeta'(-1, 3/4) (negated).

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

0,2

LINKS

Table of n, a(n) for n=0..105.

J. Miller and V. Adamchik, Derivatives of the Hurwitz Zeta Function for Rational Arguments, Journal of Computational and Applied Mathematics 100 (1998) 201-206. [contains a large number of typos]

Eric Weisstein's World of Mathematics, Hurwitz Zeta Function, formula 24.

FORMULA

Equals Pi/32 - PolyGamma(1, 1/4)/(32*Pi) - Zeta'(-1)/8.

A324995 + A324996 = -Zeta'(-1)/4.

EXAMPLE

-0.05221258304514832888285615658675114937051199095455909460693510398326...

MAPLE

evalf(Zeta(1, -1, 3/4), 120);

evalf(Pi/32 - Psi(1, 1/4)/(32*Pi) - Zeta(1, -1)/8, 120);

MATHEMATICA

RealDigits[Derivative[1, 0][Zeta][-1, 3/4], 10, 120][[1]]

N[With[{k=1}, (4^k-1) * BernoulliB[2*k] * Pi/4^(k+1)/k + (4^(k-1) - 1) * BernoulliB[2*k] * Log[2]/k/2^(4*k-1) + (-1)^k*PolyGamma[2*k-1, 1/4] / 4 / (8*Pi)^(2*k-1) - (4^k-2)*Zeta'[1-2*k]/2^(4*k)], 120]

PROG

(PARI) zetahurwitz'(-1, 3/4) \\ Michel Marcus, Mar 24 2019

CROSSREFS

Cf. A084448, A240966, A324995.

Sequence in context: A021661 A324187 A219120 * A359426 A058841 A129165

Adjacent sequences: A324993 A324994 A324995 * A324997 A324998 A324999

KEYWORD

nonn,cons

AUTHOR

Vaclav Kotesovec, Mar 23 2019

STATUS

approved