A262383 Denominators of a semi-convergent series leading to the first Stieltjes constant gamma_1.

12, 720, 15120, 11200, 332640, 908107200, 4324320, 2940537600, 175991175360, 512143632000, 1427794368, 7795757249280, 107084577600, 279490747536000, 200143324310529600, 1178332991611776000, 157531148611200, 906996615309386784000, 5828652498614400, 262872227687509440000

Offset

1,1

Comments

gamma_1 = - 1/12 + 11/720 - 137/15120 + 121/11200 - 7129/332640 + 57844301/908107200 - ..., see formulas (46)-(47) in the reference below.

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.

Formula

a(n) = denominator(-B_{2n}*H_{2n-1}/(2n)), where B_n and H_n are Bernoulli and harmonic numbers respectively.

a(n) = denominator(Zeta(1 - 2*n)*(Psi(2*n) + gamma)), where gamma is Euler's gamma. - Peter Luschny, Apr 19 2018

Example

Denominators of -1/12, 11/720, -137/15120, 121/11200, -7129/332640, 57844301/908107200, ...

Maple

a := n -> denom(Zeta(1 - 2*n)*(Psi(2*n) + gamma)):
seq(a(n), n=1..20); # Peter Luschny, Apr 19 2018

Mathematica

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

Prog

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

Crossrefs

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

Sequence in context: A171105 A215686 A277691 * A002196 A141421 A000909

Adjacent sequences: A262380 A262381 A262382 * A262384 A262385 A262386

Keyword

nonn,frac

Author

Iaroslav V. Blagouchine, Sep 20 2015

Status

approved