A262382 Numerators of a semi-convergent series leading to the first Stieltjes constant gamma_1.

-1, 11, -137, 121, -7129, 57844301, -1145993, 4325053069, -1848652896341, 48069674759189, -1464950131199, 105020512675255609, -22404210159235777, 1060366791013567384441, -15899753637685210768473787, 2241672100026760127622163469, -8138835628210212414423299

Offset

1,2

Comments

gamma_1 = - 1/12 + 11/720 - 137/15120 + 121/11200 - 7129/332640 + 57844301/908107200 - ..., see formulas (46)-(47) in the reference below.

Links

G. C. Greubel, Table of n, a(n) for n = 1..259

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_{2n-1}/(2n)), where B_n and H_n are Bernoulli and harmonic numbers respectively.

a(n) = numerator(Zeta(1 - 2*n)*(Psi(2*n) + gamma)), where gamma is Euler's gamma. - Peter Luschny, Apr 19 2018

Example

Numerators of -1/12, 11/720, -137/15120, 121/11200, -7129/332640, 57844301/908107200, ...

Maple

a := n -> numer(Zeta(1 - 2*n)*(Psi(2*n) + gamma)):
seq(a(n), n=1..16); # Peter Luschny, Apr 19 2018

Mathematica

a[n_] := Numerator[-BernoulliB[2*n]*HarmonicNumber[2*n - 1]/(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*n)); \\ Michel Marcus, Sep 23 2015

Crossrefs

Cf. A001620, A002206, A195189, A075266, A262235, A001067, A006953, A082633, A262383 (denominators of this series), A086279, A086280, A262387.

Sequence in context: A057718 A123800 A233258 * A142895 A232180 A201111

Adjacent sequences: A262379 A262380 A262381 * A262383 A262384 A262385

Keyword

frac,sign

Author

Iaroslav V. Blagouchine, Sep 20 2015

Status

approved