A350670 Denominators of Sum_{j=0..n} 1/(2*j+1), for n >= 0.
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
Links
- Hugo Pfoertner, Table of n, a(n) for n = 0..100
- 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.
- Comparison to A025547 using Plot 2.
Programs
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Magma
[Denominator((2*HarmonicNumber(2*n+2) - HarmonicNumber(n+1))): n in [0..40]]; // G. C. Greubel, Jul 24 2023
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Mathematica
With[{H=HarmonicNumber}, Table[Denominator[2*H[2n+2] -H[n+1]], {n,0,50}]] (* G. C. Greubel, Jul 24 2023 *) -
PARI
a(n) = denominator(sum(j=0, n, 1/(2*j+1))); \\ Michel Marcus, Mar 16 2022
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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
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
Comments