cp's OEIS Frontend

A335955 a(n) = (4^n*(Z(-n, 1/4) - Z(-n, 3/4)) + Z(-n, 1)*(2^(n+1)-1))*A171977(n+1), where Z(n, c) is the Hurwitz zeta function.

%I A335955 #16 Feb 16 2025 08:34:00
%S A335955 0,-1,-1,1,5,-1,-61,17,1385,-31,-50521,691,2702765,-5461,-199360981,
%T A335955 929569,19391512145,-3202291,-2404879675441,221930581,370371188237525,
%U A335955 -4722116521,-69348874393137901,968383680827,15514534163557086905,-14717667114151,-4087072509293123892361
%N A335955 a(n) = (4^n*(Z(-n, 1/4) - Z(-n, 3/4)) + Z(-n, 1)*(2^(n+1)-1))*A171977(n+1), where Z(n, c) is the Hurwitz zeta function.
%H A335955 N. D. Elkies, <a href="https://arxiv.org/abs/math/0101168">On the sums Sum((4k+1)^(-n),k,-inf,+inf)</a>, arXiv:math/0101168 [math.CA], 2001.
%H A335955 N. D. Elkies, <a href="http://www.jstor.org/stable/3647742">On the sums Sum_{k = -infinity .. infinity} (4k+1)^(-n)</a>, Amer. Math. Monthly, 110 (No. 7, 2003), 561-573.
%H A335955 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/FavardConstants.html">Favard Constants</a>
%F A335955 A002425 interleaved with A028296.
%F A335955 |Numerator(a(n)/n!)| = A050970(n+1) for n >= 1.
%F A335955 a(n) = 2*(4^n*(Z(-n, 1/4) - Z(-n, 3/4)) + Z(-n,1)*A335954(n+1)) where Z(n, c) is the Hurwitz zeta function.
%p A335955 HZeta := (s, v) -> Zeta(0, s, v):
%p A335955 a := s -> (4^s*(HZeta(-s,1/4) - HZeta(-s,3/4)) + HZeta(-s,1)*(2^(s+1)-1))* 2^padic[ordp](2*(s+1),2): seq(a(n), n = 0..28);
%t A335955 a[n_] := 2^(IntegerExponent[n + 1, 2] + 1) (4^n (HurwitzZeta[-n, 1/4] - HurwitzZeta[-n, 3/4]) + HurwitzZeta[-n, 1] (2^(n + 1) - 1));
%t A335955 Table[FullSimplify[a[n]], {n, 0, 26}]
%Y A335955 Cf. A002425, A028296, A050970, A171977, A335954.
%K A335955 sign
%O A335955 0,5
%A A335955 _Peter Luschny_, Jul 20 2020