OFFSET
1,3
COMMENTS
gamma_3 = + 1/120 - 17/1008 + 967/28800 - 4523/49896 + 33735311/101088000 - ..., see formulas (46)-(47) in the reference below.
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..225
Iaroslav V. Blagouchine, Expansions of generalized Euler's constants into the series of polynomials in 1/pi^2 and into the formal enveloping series with rational coefficients only, Journal of Number Theory (Elsevier), vol. 158, pp. 365-396, 2016. arXiv version, arXiv:1501.00740 [math.NT], 2015.
FORMULA
a(n) = numerator(-B_{2n}*(H^3_{2n-1}-3*H_{2n-1}*H^(2)_{2n-1}+2*H^(3)_{2n-1})/(2n)), where B_n, H_n and H^(k)_n are Bernoulli, harmonic and generalized harmonic numbers respectively.
EXAMPLE
Numerators of -0/1, 1/120, -17/1008, 967/28800, -4523/49896, 33735311/101088000, ...
MATHEMATICA
a[n_] := Numerator[-BernoulliB[2*n]*(HarmonicNumber[2*n - 1]^3 - 3*HarmonicNumber[2*n - 1]*HarmonicNumber[2*n - 1, 2] + 2*HarmonicNumber[2*n - 1, 3])/(2*n)]; Table[a[n], {n, 1, 20}]
PROG
(PARI) a(n) = numerator(-bernfrac(2*n)*(sum(k=1, 2*n-1, 1/k)^3 -3*sum(k=1, 2*n-1, 1/k)*sum(k=1, 2*n-1, 1/k^2) + 2*sum(k=1, 2*n-1, 1/k^3))/(2*n));
CROSSREFS
KEYWORD
frac,sign
AUTHOR
Iaroslav V. Blagouchine, Sep 20 2015
STATUS
approved