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A350669 Numerators of Sum_{j=0..n} 1/(2*j+1), for n >= 0.

%I A350669 #27 Aug 13 2023 17:33:16
%S A350669 1,4,23,176,563,6508,88069,91072,1593269,31037876,31730711,744355888,
%T A350669 3788707301,11552032628,340028535787,10686452707072,10823198495797,
%U A350669 10952130239452,409741429887649,414022624965424,17141894231615609,743947082888833412,750488463554668427,35567319917031991744,250947670863258378883,252846595191840484708,13497714685925233086599
%N A350669 Numerators of Sum_{j=0..n} 1/(2*j+1), for n >= 0.
%C A350669 For the denominators see A350670.
%C A350669 This sequence coincides with A025550(n+1), for n = 0, 1, ..., 37. See the comments there.
%C A350669 Thanks to _Ralf Steiner_ for sending me a paper where this and similar sums appear.
%H A350669 Hugo Pfoertner, <a href="/A350669/b350669.txt">Table of n, a(n) for n = 0..100</a>
%H A350669 Abramowitz and I. 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 A350669 <a href="https://oeis.org/plot2a?graph=1&amp;name1=A350669&amp;name2=A025550&amp;tform1=log%20base%2010&amp;tform2=log%20base%2010&amp;shift=1&amp;radiop1=matp&amp;drawpoints=true">Comparison to A025550 using Plot 2</a>.
%F A350669 a(n) = numerator((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 A350669 a(n) = (1/2) * numerator 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 A350669 With[{H=HarmonicNumber}, Table[Numerator[2*H[2*n+2] -H[n+1]]/2 , {n,0,50}]] (* _G. C. Greubel_, Jul 24 2023 *)
%o A350669 (PARI) a(n) = numerator(sum(j=0, n, 1/(2*j+1))); \\ _Michel Marcus_, Mar 16 2022
%o A350669 (Magma) [Numerator((2*HarmonicNumber(2*n+2) - HarmonicNumber(n+1)))/2: n in [0..40]]; // _G. C. Greubel_, Jul 24 2023
%o A350669 (SageMath) [numerator(2*harmonic_number(2*n+2,1) - harmonic_number(n+1,1))/2 for n in range(41)] # _G. C. Greubel_, Jul 24 2023
%Y A350669 Cf. A001620, A025547, A025550, A111877 (denominators), A350670.
%K A350669 nonn,frac,easy
%O A350669 0,2
%A A350669 _Wolfdieter Lang_, Mar 16 2022