A397636 - OEIS

%I #7 Jul 03 2026 08:21:50

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

%T 6,5,0,7,1,0,1,1,3,5,5,7,2,6,6,3,3,8,0,5,5,0,9,8,5,6,8,9,0,7,8,6,7,6,

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

%N Decimal expansion of Sum_{k>=1} H(k)^3*H(k,2)/k^2, where H(k) = A001008(k)/A002805(k) is the k-th harmonic number, and H(k,2) = A007406(k)/A007407(k) is the k-th generalized harmonic number of order 2.

%H Cornel Ioan Vălean, <a href="https://doi.org/10.1007/978-3-030-02462-8">(Almost) Impossible Integrals, Sums, and Series</a>, Springer International Publishing, 2019, section 4.43, pp. 302-303, eq. (4.66), section 5.43, p. 324, section 6.43, pp. 470-479.

%H <a href="/index/Ha#harmonic">Index entries for sequences related to harmonic numbers</a>.

%F Equals 83*zeta(7)/16 - 5*zeta(2)*zeta(5)/2 + 27*zeta(3)*zeta(4)/2.

%e 18.53030794936785681409856255962955650710113557266338...

%t RealDigits[83*Zeta[7]/16 - 5*Zeta[2]*Zeta[5]/2 + 27*Zeta[3]*Zeta[4]/2, 10, 120][[1]]

%o (PARI) 83*zeta(7)/16 - 5*zeta(2)*zeta(5)/2 + 27*zeta(3)*zeta(4)/2

%Y Cf. A001008, A002805, A007406, A007407.

%Y Cf. A002117, A013661, A013662, A013663, A013665.

%K nonn,cons

%O 2,2

%A _Amiram Eldar_, Jul 03 2026