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A350670 Denominators of Sum_{j=0..n} 1/(2*j+1), for n >= 0.

%I A350670 #33 May 15 2024 11:04:22
%S A350670 1,3,15,105,315,3465,45045,45045,765765,14549535,14549535,334639305,
%T A350670 1673196525,5019589575,145568097675,4512611027925,4512611027925,
%U A350670 4512611027925,166966608033225,166966608033225,6845630929362225,294362129962575675,294362129962575675,13835020108241056725,96845140757687397075,96845140757687397075,5132792460157432044975
%N A350670 Denominators of Sum_{j=0..n} 1/(2*j+1), for n >= 0.
%C A350670 For the numerators see A350669.
%C A350670 This sequence coincides with A025547(n+1), for n = 0, 1, ..., 37. See the comments there.
%C A350670 Thanks to _Ralf Steiner_ for sending me a paper where this and similar sums appear.
%H A350670 Hugo Pfoertner, <a href="/A350670/b350670.txt">Table of n, a(n) for n = 0..100</a>
%H A350670 Milton Abramowitz and Irene A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP?Res=150&amp;Page=258">Handbook of Mathematical Functions. p. 258</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy], p. 258.
%H A350670 Yue-Wu Li and Feng Qi, <a href="https://doi.org/10.3390/axioms13050317">A New Closed-Form Formula of the Gauss Hypergeometric Function at Specific Arguments</a>, Axioms (2024) Vol. 13, Art. No. 317. See p. 11 of 24.
%H A350670 <a href="https://oeis.org/plot2a?graph=1&amp;name1=A350670&amp;name2=A025547&amp;tform1=log%20base%2010&amp;tform2=log%20base%2010&amp;shift=1&amp;radiop1=matp&amp;drawpoints=true">Comparison to A025547 using Plot 2</a>.
%F A350670 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.
%F A350670 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
%t A350670 With[{H=HarmonicNumber}, Table[Denominator[2*H[2n+2] -H[n+1]], {n,0,50}]] (* _G. C. Greubel_, Jul 24 2023 *)
%o A350670 (PARI) a(n) = denominator(sum(j=0, n, 1/(2*j+1))); \\ _Michel Marcus_, Mar 16 2022
%o A350670 (Magma) [Denominator((2*HarmonicNumber(2*n+2) - HarmonicNumber(n+1))): n in [0..40]]; // _G. C. Greubel_, Jul 24 2023
%o A350670 (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
%Y A350670 Cf. A001620, A025547, A025550, A350669 (numerators).
%K A350670 nonn,frac,easy
%O A350670 0,2
%A A350670 _Wolfdieter Lang_, Mar 16 2022