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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
1, 8, 256, 17408, 2031616, 362283008, 91620376576, 31191159799808, 13753735117275136, 7625476699018231808, 5192022022552652087296, 4258996468871236847403008, 4142655008190840426050093056, 4714505177821257067736657297408, 6206008749802659037752564348092416
OFFSET
0,2
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
KEYWORD
nonn
AUTHOR
Peter Luschny, Mar 11 2015
STATUS
approved