A393116 Decimal expansion of Sum_{k>=1} (O(k)/k)^2, where O(k) = A350669(k)/A350670(k) is the k-th odd harmonic number (or harmonic number of the second kind).

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

Offset

1,1

Comments

The closed-form expression for this sum was found by De Doelder (1991).

Links

Table of n, a(n) for n=1..105.

David Borwein and Jonathan M. Borwein, On an intriguing integral and some series related to zeta(4), Proceedings of the American Mathematical Society, Vol. 123, No. 4 (1995), pp. 1191-1198.

P. J. De Doelder, On some series containing psi(x)-psi(y) and (psi(x)-psi(y))^2 for certain values of x and y, Journal of Computational and Applied Mathematics, Vol. 37, No. 1-3 (1991), pp. 125-141. See p. 138, eq. (22).

Formula

Equals Pi^4/32.

Equals 45*zeta(4)/16.

Example

3.04403409481257616363876039652203472655398705227141...

Mathematica

RealDigits[Pi^4/32, 10, 120][[1]]

Prog

(PARI) Pi^4/32

Crossrefs

Cf. A350669, A350670.

Cf. A013662, A092425.

Related constants: A218505, A393115, A393117, A393118, A393119, A393120, A393143.

Sequence in context: A099925 A351002 A201563 * A387770 A016644 A240966

Adjacent sequences: A393113 A393114 A393115 * A393117 A393118 A393119

Keyword

nonn,cons

Author

Amiram Eldar, Feb 02 2026

Status

approved