A253165 a(n) = (-1)^n*2^(6*n+3)*(zeta(-2*n-1,1/2) - zeta(-2*n-1,1)), where zeta(a,z) is the generalized Riemann zeta function.

1, 8, 256, 17408, 2031616, 362283008, 91620376576, 31191159799808, 13753735117275136, 7625476699018231808, 5192022022552652087296, 4258996468871236847403008, 4142655008190840426050093056, 4714505177821257067736657297408, 6206008749802659037752564348092416

Offset

0,2

Links

Table of n, a(n) for n=0..14.

Formula

a(n) = (-1)^n*2^(4*n+1)*(E(2*n+1,1/2)-E(2*n+1,0)), where E(n,x) are the Euler polynomials.

a(n) = A000825(2*n+1).

a(n) = A000828(2*n+1).

a(n) = A000831(2*n+1)/2.

a(n) = A012393(n+1)/2.

G.f.: S(0), where S(k)= 1 - 4*x*(k+1)*(k+2)/(4*x*(k+1)*(k+2) - 1/S(k+1)); (continued fraction). - Sergei N. Gladkovskii, May 28 2015

a(n) ~ (2*n+1)! * 2^(4*n+3) / Pi^(2*n+2). - Vaclav Kotesovec, Jun 01 2015

Maple

a := n -> (-1)^n*2^(6*n+3)*(Zeta(0, -2*n-1, 1/2)-Zeta(0, -2*n-1, 1)):
seq(a(n), n=0..14);

Mathematica

f[n_] := (-1)^n*2^(6 n + 3) (Zeta[-2 n - 1, 1/2] - Zeta[-2 n - 1, 1]); Array[f, 15, 0] (* Robert G. Wilson v, Mar 11 2015 *)
max = 20; Clear[g]; g[max + 2] = 1; g[k_] := g[k] = 1 - 4*x*(k+1)*(k+2)/(4*x*(k+1)*(k+2) - 1/g[k+1]); gf = g[0]; CoefficientList[Series[gf, {x, 0, max}], x] (* Vaclav Kotesovec, Jun 01 2015, after Sergei N. Gladkovskii *)

Crossrefs

Cf. A000825, A000828, A000831, A012393.

Sequence in context: A010044 A374283 A024290 * A140338 A140339 A103488

Adjacent sequences: A253162 A253163 A253164 * A253166 A253167 A253168

Keyword

nonn

Author

Peter Luschny, Mar 11 2015

Status

approved