A397631 - OEIS

A397631

Decimal expansion of Sum_{k>=1} H(k)*H(k,4)/k^2, where H(k) = A001008(k)/A002805(k) is the k-th harmonic number, and H(k,4) = A007410(k)/A007480(k) is the k-th harmonic number of order 4.

%I #7 Jul 03 2026 08:20:41

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

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

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

%N Decimal expansion of Sum_{k>=1} H(k)*H(k,4)/k^2, where H(k) = A001008(k)/A002805(k) is the k-th harmonic number, and H(k,4) = A007410(k)/A007480(k) is the k-th harmonic number of order 4.

%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. (4.60), 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 9*zeta(2)*zeta(5)/2 - 3*zeta(3)*zeta(4)/2 - 51*zeta(7)/16.

%e 2.509915558377422610407917096386899377535040737104199...

%t RealDigits[9*Zeta[2]*Zeta[5]/2 - 3*Zeta[3]*Zeta[4]/2 - 51*Zeta[7]/16, 10, 120][[1]]

%o (PARI) 9*zeta(2)*zeta(5)/2 - 3*zeta(3)*zeta(4)/2 - 51*zeta(7)/16

%Y Cf. A001008, A002805, A007410, A007480.

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

%K nonn,cons

%O 1,1

%A _Amiram Eldar_, Jul 03 2026