A262384 Numerators of a semi-convergent series leading to the second Stieltjes constant gamma_2.

0, -1, 5, -469, 6515, -131672123, 63427, -47800416479, 15112153995391, -29632323552377537, 4843119962464267, -1882558877249847563479, 2432942522372150087, -2768809380553055597986831, 334463513629004852735064113, -1125061940756859461946444233539, 333807583501528759350875247323

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

The sequence of denominators is A262385.

Cf. A001067, A001620, A002206, A006953, A075266, A082633, A086279, A086280, A195189, A262235, A262382, A262383, A262386, A262387.

Sequence in context: A182403 A125534 A320958 * A024071 A321939 A198978

Adjacent sequences: A262381 A262382 A262383 * A262385 A262386 A262387

Keyword

frac,sign

Author

Iaroslav V. Blagouchine, Sep 20 2015

Status

approved