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

Author

Peter Luschny, Jul 20 2020

Keywords

• sign

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

Crossrefs

Cf. A002425, A028296, A050970, A171977, A335954.

Programs

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);

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}]

Formula

A002425 interleaved with A028296.

|Numerator(a(n)/n!)| = A050970(n+1) for n >= 1.

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.