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.
0, -1, -1, 1, 5, -1, -61, 17, 1385, -31, -50521, 691, 2702765, -5461, -199360981, 929569, 19391512145, -3202291, -2404879675441, 221930581, 370371188237525, -4722116521, -69348874393137901, 968383680827, 15514534163557086905, -14717667114151, -4087072509293123892361
Offset: 0
Keywords
Links
- N. D. Elkies, On the sums Sum((4k+1)^(-n),k,-inf,+inf), arXiv:math/0101168 [math.CA], 2001.
- N. D. Elkies, On the sums Sum_{k = -infinity .. infinity} (4k+1)^(-n), Amer. Math. Monthly, 110 (No. 7, 2003), 561-573.
- Eric Weisstein's World of Mathematics, Favard Constants
Programs
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Maple
HZeta := (s, v) -> Zeta(0, s, v): 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);
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Mathematica
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)); Table[FullSimplify[a[n]], {n, 0, 26}]