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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.

%I A253165 #11 Jun 01 2015 08:32:33
%S A253165 1,8,256,17408,2031616,362283008,91620376576,31191159799808,
%T A253165 13753735117275136,7625476699018231808,5192022022552652087296,
%U A253165 4258996468871236847403008,4142655008190840426050093056,4714505177821257067736657297408,6206008749802659037752564348092416
%N 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.
%F A253165 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.
%F A253165 a(n) = A000825(2*n+1).
%F A253165 a(n) = A000828(2*n+1).
%F A253165 a(n) = A000831(2*n+1)/2.
%F A253165 a(n) = A012393(n+1)/2.
%F A253165 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
%F A253165 a(n) ~ (2*n+1)! * 2^(4*n+3) / Pi^(2*n+2). - _Vaclav Kotesovec_, Jun 01 2015
%p A253165 a := n -> (-1)^n*2^(6*n+3)*(Zeta(0,-2*n-1,1/2)-Zeta(0,-2*n-1, 1)):
%p A253165 seq(a(n), n=0..14);
%t A253165 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 *)
%t A253165 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_ *)
%Y A253165 Cf. A000825, A000828, A000831, A012393.
%K A253165 nonn
%O A253165 0,2
%A A253165 _Peter Luschny_, Mar 11 2015