OFFSET
1,2
REFERENCES
Ali Shadhar Olaikhan, An Introduction to the Harmonic Series and Logarithmic Integrals, 2021, p. 284, eq. (4.209).
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Philippe Flajolet and Bruno Salvy, Euler Sums and Contour Integral Representations, Experimental Mathematics 7:1 (1998), pp. 15-35. See page 23.
FORMULA
Equals 7*Pi^4/360 = (7/4)*A013662.
From Peter Bala, Jul 27 2025: (Start)
Series acceleration formula:
Let s(n) = Sum_{k = 1..n} H(k,2)/k^2 and S(n) = Sum_{k = 1..n} (-1)^(n+k)*binomial(n, k)*binomial(n+k, k)*s(n+k). It appears that S(n) converges much more rapidly to 7*Pi^4/360 than s(n).
For example, s(50) = 1.8(61...) is only correct to 2 decimal digits, while S(50) = 1.89406565899449183515 30064689470(06...) is correct to 32 decimal digits. (End)
Equals Sum_{k>=1} H(2*k) * AH(2*k) / k^2, where H(k) = A001008(k)/A002805(k) is the k-th harmonic number, and AH(K) = A058313(k)/A058312(k) is the k-th alternating harmonic (or skew-harmonic) number (Olaikhan, 2021). - Amiram Eldar, Feb 03 2026
EXAMPLE
1.894065658994491835153006468947043829855814165857772088445208377027211...
MATHEMATICA
RealDigits[7/4*Zeta[4], 10, 100] // First
PROG
(PARI) 7*zeta(4)/4 \\ Michel Marcus, Jul 04 2014
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Jean-François Alcover, Jul 04 2014
STATUS
approved
