A397204 Decimal expansion of Sum_{k>=1} AH(k)/(2*k+1)^4, where AH(k) = A058313(k)/A058312(k) is the k-th alternating harmonic (or skew-harmonic) number.

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

Offset

0,3

Links

Table of n, a(n) for n=0..104.

Cornel Ioan Vălean, More (Almost) Impossible Integrals, Sums, and Series, Springer Cham, 2023. See section 4.22, p. 426.

Index entries for sequences related to harmonic numbers.

Formula

Equals Pi^3*G/16 + 15*log(2)*zeta(4)/16 - Pi^5/192 - 31*zeta(5)/8 + Pi*polygamma(3, 1/4)/1536, where G is Catalan's constant (A006752), and polygamma(3, 1/4) = psi'''(1/4) is the third derivative of the digamma function at 1/4 (A390074).

Example

0.013699300049135853225210474837981346031785724130521...

Mathematica

RealDigits[Pi^3*Catalan/16 + 15*Log[2]*Zeta[4]/16 - Pi^5/192 - 31*Zeta[5]/8 + Pi*PolyGamma[3, 1/4]/1536, 10, 120, -1][[1]]

Prog

(PARI) Pi^3*Catalan/16 + 15*log(2)*zeta(4)/16 - Pi^5/192 - 31*zeta(5)/8 + Pi*psi'''(1/4)/1536

Crossrefs

Cf. A058312, A058313.

Cf. A002162, A013662, A013663, A006752, A390074.

Sum_{k>=1} AH(k)/(2*k+1)^m: A397202 (m=2), A397203 (m=3), this constant (m=4).

Sequence in context: A000748 A057083 A325738 * A198373 A331065 A160178

Adjacent sequences: A397201 A397202 A397203 * A397205 A397208 A397209

Keyword

nonn,cons,new

Author

Amiram Eldar, Jun 18 2026

Status

approved