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A262386 Numerators of a semi-convergent series leading to the third Stieltjes constant gamma_3. 3
0, 1, -17, 967, -4523, 33735311, -9301169, 127021899032857, -3546529522734769, 5633317707758173, -1935081812850766373, 779950247074296817622891, -1261508681536108282229, 350992098387568751020053498509, -17302487974885784968377519342317, 26213945071317075538702463006927083 (list; graph; refs; listen; history; text; internal format)
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

Table of n, a(n) for n=1..16.

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) = -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

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Last modified February 21 22:56 EST 2018. Contains 299427 sequences. (Running on oeis4.)