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A156201 Numerator of Euler(n, 1/8).

%I A156201 #30 Aug 24 2022 01:50:04
%S A156201 1,-3,-7,117,497,-15123,-95767,4116837,34741217,-1921996323,
%T A156201 -20273087527,1370953667157,17352768515537,-1386843017916723,
%U A156201 -20479521468959287,1888542637550347077,31872138933891307457,-3331009898404800736323,-63243057486503656319047
%N A156201 Numerator of Euler(n, 1/8).
%H A156201 T. D. Noe, <a href="/A156201/b156201.txt">Table of n, a(n) for n = 0..100</a>
%F A156201 a(n) = Im(2*i*(1+Sum_{j=0..n} (binomial(n,j)*Li_{-j}(i)*4^j))). - _Peter Luschny_, Apr 29 2013
%F A156201 G.f.: conjecture T(0)/(1+3*x), where T(k) = 1 - 16*x^2*(k+1)^2/(16*x^2*(k+1)^2 + (1+3*x)^2/T(k+1) ); (continued fraction). - _Sergei N. Gladkovskii_, Nov 12 2013
%F A156201 a(n) = (-4)^n*skp(n, 3/4), where skp(n,x) are the Swiss-Knife polynomials A153641. - _Peter Luschny_, Apr 19 2014
%F A156201 a(n) = 2^(4*n+1)*(zeta(-n,1/16)-zeta(-n, 9/16)), where zeta(a, z) is the generalized Riemann zeta function. - _Peter Luschny_, Mar 11 2015
%F A156201 From _Emanuele Munarini_, Aug 22 2022: (Start)
%F A156201 E.g.f.: (2*e^t)/(e^(8*t)+1).
%F A156201 E.g.f. for the sequence of the absolute values: (cos(3*t)+sin(3*t))/cos(4*t).
%F A156201 |a(2*n)| = Sum_{k=0..n} binomial(2*n,2*k) (-1)^k 4^(2*n-2*k) 3^(2*k) |E(2*n-2k)|
%F A156201 |a(2*n+1)| = Sum_{k=0..n} binomial(2*n+1,2*k+1) (-1)^k 4^(2*n-2*k) 3^(2*k+1) |E(2*n-2*k)|
%F A156201 where the E(n)'s are the Euler numbers (A122045).
%F A156201 (End)
%p A156201 p := proc(n) local j; 2*I*(1+add(binomial(n,j)*polylog(-j,I)*4^j, j=0..n)) end:  A156201 := n -> Im(p(n));
%p A156201 seq(A156201(i), i=0..10);  # _Peter Luschny_, Apr 29 2013
%t A156201 Table[EulerE[n, 1/8] // Numerator, {n, 0, 18}] (* _Jean-François Alcover_, Apr 30 2013 *)
%Y A156201 For denominators see A001018. Cf. A000813.
%K A156201 sign,frac
%O A156201 0,2
%A A156201 _N. J. A. Sloane_, Nov 07 2009