login
Numerators of a semi-convergent series leading to the third Stieltjes constant gamma_3.
6

%I #31 Aug 29 2018 02:51:24

%S 0,1,-17,967,-4523,33735311,-9301169,127021899032857,

%T -3546529522734769,5633317707758173,-1935081812850766373,

%U 779950247074296817622891,-1261508681536108282229,350992098387568751020053498509,-17302487974885784968377519342317,26213945071317075538702463006927083

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

%C gamma_3 = + 1/120 - 17/1008 + 967/28800 - 4523/49896 + 33735311/101088000 - ..., see formulas (46)-(47) in the reference below.

%H G. C. Greubel, <a href="/A262386/b262386.txt">Table of n, a(n) for n = 1..225</a>

%H Iaroslav V. Blagouchine, <a href="http://dx.doi.org/10.1016/j.jnt.2015.06.012">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</a>, Journal of Number Theory (Elsevier), vol. 158, pp. 365-396, 2016. <a href="http://arxiv.org/abs/1501.00740">arXiv version</a>, arXiv:1501.00740 [math.NT], 2015.

%F 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.

%e Numerators of -0/1, 1/120, -17/1008, 967/28800, -4523/49896, 33735311/101088000, ...

%t 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}]

%o (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));

%Y The sequence of denominators is A262387.

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

%K frac,sign

%O 1,3

%A _Iaroslav V. Blagouchine_, Sep 20 2015