A262385 Denominators of a semi-convergent series leading to the second Stieltjes constant gamma_2.

1, 60, 336, 21600, 133056, 825552000, 89100, 11435424000, 483113030400, 101889627840000, 1471926193920, 42280119968486400, 3425059028160, 209827678712652000, 1184296360402995840, 163066081742403840000, 1749151741873536000, 20373357051590182072392960000

Offset: 1

Author

Iaroslav V. Blagouchine, Sep 20 2015

Keywords

• nonn
• frac

Comments

gamma_2 = - 1/60 + 5/336 - 469/21600 + 6515/133056 - 131672123/825552000 + ..., see formulas (46)-(47) in the reference below.

Examples

			Denominators of 0/1, -1/60, 5/336, -469/21600, 6515/133056, -131672123/825552000, ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..500
• 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

Cf. A001620, A002206, A195189, A075266, A262235, A001067, A006953, A082633, A262382, A262383, A086279, A262384 (numerators of this series), A086280, A262386, A262387.

Programs

Maple

a := n -> denom(-Zeta(1 - 2*n)*(Psi(1, 2*n) + (Psi(0,2*n) + gamma)^2 - (Pi^2)/6)):
seq(a(n), n=1..18); # Peter Luschny, Apr 19 2018

Mathematica

a[n_] := Denominator[BernoulliB[2*n]*(HarmonicNumber[2*n - 1]^2 - HarmonicNumber[2*n - 1, 2])/(2*n)]; Table[a[n], {n, 1, 20}]

PARI

a(n) = denominator(bernfrac(2*n)*(sum(k=1,2*n-1,1/k)^2 - sum(k=1,2*n-1,1/k^2))/(2*n)); \\ Michel Marcus, Sep 23 2015

Formula

a(n) = denominator(B_{2n}*(H^2_{2n-1}-H^(2)_{2n-1})/(2n)), where B_n, H_n and H^(k)_n are Bernoulli, harmonic and generalized harmonic numbers respectively.

a(n) = denominator(-Zeta(1 - 2*n)*(Psi(1,2*n) + (Psi(0,2*n) + gamma)^2 - (Pi^2)/6)), where gamma is Euler's gamma and Psi is the digamma function. - Peter Luschny, Apr 19 2018