A394645 - OEIS

A394645

Decimal expansion of Sum_{k>=1} H(k)^2*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.

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

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OFFSET

1,1

REFERENCES

Ali Shadhar Olaikhan, An Introduction to the Harmonic Series and Logarithmic Integrals, 2021, p. 228, eq. (4.119).

LINKS

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

Ali Shadhar Olaikhan, An Introduction to the Harmonic Series and Logarithmic Integrals, 2nd ed., 2023, section 4.2.15, pp. 295-297, eq. (4.61).

Cornel Ioan Vălean, (Almost) Impossible Integrals, Sums, and Series, Springer International Publishing, 2019, section 4.30, p. 296, eq. (4.38), section 5.30, p. 321, section 6.30, pp. 421-423.

Index entries for sequences related to harmonic numbers.

FORMULA

Equals 41*zeta(6)/12 + 2*zeta(3)^2.

EXAMPLE

6.365803725314136361851973084399620791365347235806693...

MATHEMATICA

RealDigits[41*Zeta[6]/12 + 2*Zeta[3]^2, 10, 120][[1]]

PROG

(PARI) 41*zeta(6)/12 + 2*zeta(3)^2

CROSSREFS

Cf. A001008, A002805, A007406, A007407.

Cf. A002117, A013664.