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A352395 Denominator of Sum_{k=0..n} (-1)^k / (2*k+1).

%I A352395 #28 Aug 13 2023 17:33:24
%S A352395 1,3,15,105,315,3465,45045,45045,765765,14549535,14549535,334639305,
%T A352395 1673196525,5019589575,145568097675,4512611027925,4512611027925,
%U A352395 4512611027925,166966608033225,166966608033225,6845630929362225,294362129962575675,294362129962575675,13835020108241056725
%N A352395 Denominator of Sum_{k=0..n} (-1)^k / (2*k+1).
%C A352395 This is not the sequence A025547(n+1)_{n>=0}, because a(32) = 1420993851085122917681925 but A025547(33) = 18472920064106597929865025. Hence it is also not the sequence A350670.
%C A352395 The alternating sum Sum_{k=0..n} (-1)^k/(2*k+1) = (Psi(n + 3/2) - Psi((2*n - (-1)^n)/4 + 1) - log(2) + Pi/2)/2, with the Digamma function Psi(z).
%C A352395 Proof by subtracting twice the negative fractions from Sum_{k=0..n} 1/(2*k+1) = A350669(n)/A350670(n) (Abramowitz-Stegun, p. 258, eq. 6.3.4.), using Sum_{j=0..k} 1/(4*j + 3) = A074637((k+1)/4)/A074638(k+1) (Abramowitz-Stegun, p. 258, eqs. 6.3.6. with 6.3.5.) and, finally, replacing in the results for the even and odd n cases the formula for Psi(3/4) = -A200134.
%H A352395 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.
%F A352395 a(n) = denominator( (Psi(n + 3/2) - Psi((2*n - (-1)^n)/4 + 1) - log(2) + Pi/2)/2), for n >= 0, with the Digamma function. See the above comment.
%F A352395 a(n) = denominator(Pi/4 + (-1)^n * (Psi((n + 5/2)/2) - Psi((n + 3/2)/2))/4). - _Vaclav Kotesovec_, May 16 2022
%t A352395 Denominator @ Accumulate @ Table[(-1)^k/(2*k + 1), {k, 0, 25}] (* _Amiram Eldar_, Apr 08 2022 *)
%o A352395 (PARI) a(n) = denominator(sum(k=0, n, (-1)^k / (2*k+1))); \\ _Michel Marcus_, Apr 07 2022
%o A352395 (Python)
%o A352395 from fractions import Fraction
%o A352395 def A352395(n): return sum(Fraction(-1 if k % 2 else 1,2*k+1) for k in range(n+1)).denominator # _Chai Wah Wu_, May 18 2022
%Y A352395 Cf. A007509 (numerators).
%Y A352395 Cf. A025547, A074637, A074638, A200134, A350669, A350670.
%K A352395 nonn,frac,easy
%O A352395 0,2
%A A352395 _Wolfdieter Lang_, Apr 06 2022