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A262387 Denominators of a semi-convergent series leading to the third Stieltjes constant gamma_3. 8
1, 120, 1008, 28800, 49896, 101088000, 5702400, 12350257920000, 43480172736000, 7075668600000, 206069667148800, 5919216795588096000, 581222138112000, 8460252005694128640000, 18991807088644406016000, 1150594272774401495040000, 33940540399314092544000, 9737059611553100811150566400000, 1290633707289706940160000, 1263402804161736165764268432000000 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

gamma_3 = + 1/120 - 17/1008 + 967/28800 - 4523/49896 + 33735311/101088000 - ..., 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, 2015.

FORMULA

a(n) = denominator(-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

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

MATHEMATICA

a[n_] := Denominator[-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) = denominator(-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

Cf. A001620, A002206, A195189, A075266, A262235, A001067, A006953, A082633, A262382, A262383, A086279, A262384, A262385, A086280, A262386 (numerators of this series).

Sequence in context: A011245 A213875 A092182 * A133119 A052777 A052765

Adjacent sequences:  A262384 A262385 A262386 * A262388 A262389 A262390

KEYWORD

nonn,frac

AUTHOR

Iaroslav V. Blagouchine, Sep 20 2015

STATUS

approved

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Last modified October 17 09:58 EDT 2018. Contains 316276 sequences. (Running on oeis4.)