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A350670
Denominators of Sum_{j=0..n} 1/(2*j+1), for n >= 0.
5
1, 3, 15, 105, 315, 3465, 45045, 45045, 765765, 14549535, 14549535, 334639305, 1673196525, 5019589575, 145568097675, 4512611027925, 4512611027925, 4512611027925, 166966608033225, 166966608033225, 6845630929362225, 294362129962575675, 294362129962575675, 13835020108241056725, 96845140757687397075, 96845140757687397075, 5132792460157432044975
OFFSET
0,2
COMMENTS
For the numerators see A350669.
This sequence coincides with A025547(n+1), for n = 0, 1, ..., 37. See the comments there.
Thanks to Ralf Steiner for sending me a paper where this and similar sums appear.
LINKS
Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions. p. 258, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy], p. 258.
Yue-Wu Li and Feng Qi, A New Closed-Form Formula of the Gauss Hypergeometric Function at Specific Arguments, Axioms (2024) Vol. 13, Art. No. 317. See p. 11 of 24.
FORMULA
a(n) = denominator((Psi(n+3/2) + gamma + 2*log(2))/2), with the Digamma function Psi(z), and the Euler-Mascheroni constant gamma = A001620. See Abramowitz-Stegun, p. 258, 6.3.4.
a(n) = denominator of ( 2*H_{2*n+2} - H_{n+1} ), where H_{n} is the n-th Harmonic number. - G. C. Greubel, Jul 24 2023
MATHEMATICA
With[{H=HarmonicNumber}, Table[Denominator[2*H[2n+2] -H[n+1]], {n, 0, 50}]] (* G. C. Greubel, Jul 24 2023 *)
PROG
(PARI) a(n) = denominator(sum(j=0, n, 1/(2*j+1))); \\ Michel Marcus, Mar 16 2022
(Magma) [Denominator((2*HarmonicNumber(2*n+2) - HarmonicNumber(n+1))): n in [0..40]]; // G. C. Greubel, Jul 24 2023
(SageMath) [denominator(2*harmonic_number(2*n+2, 1) - harmonic_number(n+1, 1)) for n in range(41)] # G. C. Greubel, Jul 24 2023
CROSSREFS
Cf. A001620, A025547, A025550, A350669 (numerators).
Sequence in context: A145624 A025547 A352395 * A376054 A220747 A088989
KEYWORD
nonn,frac,easy
AUTHOR
Wolfdieter Lang, Mar 16 2022
STATUS
approved