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
Examples
Numerators of -1/12, 11/720, -137/15120, 121/11200, -7129/332640, 57844301/908107200, ...
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.
Crossrefs
Programs
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Maple
a := n -> numer(Zeta(1 - 2*n)*(Psi(2*n) + gamma)): seq(a(n), n=1..16); # Peter Luschny, Apr 19 2018
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Mathematica
a[n_] := Numerator[-BernoulliB[2*n]*HarmonicNumber[2*n - 1]/(2*n)]; Table[a[n], {n, 1, 20}] -
PARI
a(n) = numerator(-bernfrac(2*n)*sum(k=1,2*n-1,1/k)/(2*n)); \\ Michel Marcus, Sep 23 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
Comments