A262386 Numerators of a semi-convergent series leading to the third Stieltjes constant gamma_3.

0, 1, -17, 967, -4523, 33735311, -9301169, 127021899032857, -3546529522734769, 5633317707758173, -1935081812850766373, 779950247074296817622891, -1261508681536108282229, 350992098387568751020053498509, -17302487974885784968377519342317, 26213945071317075538702463006927083

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

The sequence of denominators is A262387.

Cf. A001067, A001620, A002206, A006953, A075266, A082633, A086279, A086280, A195189, A262235, A262382, A262383, A262384, A262385.

Sequence in context: A156138 A229261 A196873 * A298302 A166188 A274548

Adjacent sequences: A262383 A262384 A262385 * A262387 A262388 A262389

Keyword

frac,sign

Author

Iaroslav V. Blagouchine, Sep 20 2015

Status

approved