OFFSET
1,3
COMMENTS
gamma_2 = - 1/60 + 5/336 - 469/21600 + 6515/133056 - 131672123/825552000 + ..., see formulas (46)-(47) in the reference below.
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..237
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^2_{2n-1}-H^(2)_{2n-1})/(2n)), where B_n, H_n and H^(k)_n are Bernoulli, harmonic and generalized harmonic numbers respectively.
a(n) = numerator(-Zeta(1 - 2*n)*(Psi(1,2*n) + (Psi(0,2*n) + gamma)^2 - (Pi^2)/6)), where gamma is Euler's gamma and Psi is the digamma function. - Peter Luschny, Apr 19 2018
EXAMPLE
Numerators of 0/1, -1/60, 5/336, -469/21600, 6515/133056, -131672123/825552000, ...
MAPLE
a := n -> numer(-Zeta(1 - 2*n)*(Psi(1, 2*n) + (Psi(0, 2*n) + gamma)^2 - (Pi^2)/6)):
seq(a(n), n=1..17); # Peter Luschny, Apr 19 2018
MATHEMATICA
a[n_] := Numerator[BernoulliB[2*n]*(HarmonicNumber[2*n - 1]^2 - HarmonicNumber[2*n - 1, 2])/(2*n)]; Table[a[n], {n, 1, 20}]
PROG
(PARI) a(n) = numerator(bernfrac(2*n)*(sum(k=1, 2*n-1, 1/k)^2 - sum(k=1, 2*n-1, 1/k^2))/(2*n)); \\ Michel Marcus, Sep 23 2015
CROSSREFS
KEYWORD
frac,sign
AUTHOR
Iaroslav V. Blagouchine, Sep 20 2015
STATUS
approved