A393120 Decimal expansion of Sum_{k>=1} (-1)^(k+1) * (AO(k)/k)^2, where AO(k) = A007509(k)/A352395(k) is the k-th alternating odd harmonic number (or skew-harmonic number of the second kind).

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

Offset

0,1

Links

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

Cornel Ioan Vălean, Another Masterpiece: The Evaluation of the Skew-Harmonic Number of the Second Kind Version of the Alternating Au-Yeung Series, ResearchGate, April 2025.

Formula

Equals 2*G^2 - log(2)*Pi*G + (3/8)*log(2)^2*zeta(2) + (135/32)*zeta(4) - 2*Pi*Im(Li_3((1+i)/2)), where G is Catalan's constant (A006752), Li_3(z) is the polylogarithm function of order 3, and i = sqrt(-1).

Example

0.963908730192273667827365737552590076416576545223573...

Mathematica

RealDigits[2*Catalan^2 - Log[2]*Pi*Catalan + (3/8)*Log[2]^2*Zeta[2] + (135/32)*Zeta[4] - 2*Pi*Im[PolyLog[3, (1 + I)/2]], 10, 120][[1]]

Prog

(PARI) 2*Catalan^2 - log(2)*Pi*Catalan + (3/8)*log(2)^2*zeta(2) + (135/32)*zeta(4) - 2*Pi*imag(polylog(3, (1+I)/2))

Crossrefs

Cf. A007509, A352395.

Cf. A002162, A013661, A013662, A006752, A355022.

Related constants: A218505, A393115, A393116, A393117, A393118, A393119, A393143.

Sequence in context: A259469 A241993 A099817 * A262701 A200280 A198363

Adjacent sequences: A393117 A393118 A393119 * A393121 A393122 A393123

Keyword

nonn,cons

Author

Amiram Eldar, Feb 02 2026

Status

approved