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%I #60 Apr 16 2022 05:15:06
%S 0,2,40,1952,177280,25866752,5535262720,1633165156352,635421069967360,
%T 315212388819402752,194181169538675507200,145435130631317935357952,
%U 130145345400688287667978240,137139396592145493713802493952
%N E.g.f.: Sum_{n >= 0} a(n) * x^(2*n) / (2*n)! = sin(x)^2 / cos(2*x).
%H R. J. Mathar, <a href="/A000816/b000816.txt">Table of n, a(n) for n = 0..200</a>
%F (1/2) * A002436(n), n > 0. - _Ralf Stephan_, Mar 09 2004
%F a(n) = 2^(2*n - 1) * A000364(n) except at n=0.
%F E.g.f.: sin(x)^2/cos(2x) = 1/Q(0) - 1/2; Q(k) = 1 + 1/(1-2*(x^2)/(2*(x^2)-(k+1)*(2k+1)/Q(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, Nov 18 2011
%F a(n) = A000819(n) unless n=0.
%F G.f.: (1/(G(0))-1)/2 where G(k) = 1 - 4*x*(k+1)^2/G(k+1); (continued fraction). - _Sergei N. Gladkovskii_, Jan 12 2013
%F G.f.: T(0)/2 - 1/2, where T(k) = 1 - 4*x*(k+1)^2/( 4*x*(k+1)^2 - 1/T(k+1) ); (continued fraction). - _Sergei N. Gladkovskii_, Oct 25 2013
%F E.g.f.: sin(x)^2/cos(2*x) = x^2/(1-2*x^2)*T(0), where T(k) = 1 - x^2*(2*k+1)*(2*k+2)/( x^2*(2*k+1)*(2*k+2) + ((k+1)*(2*k+1) - 2*x^2)*((k+2)*(2*k+3) - 2*x^2)/T(k+1) ); (continued fraction). - _Sergei N. Gladkovskii_, Oct 25 2013
%F From _Artur Jasinski_, Mar 21 2022: (Start)
%F For n > 0:
%F a(n) = Pi^(2*n-1)*(-Psi(2*n,1/4) - (4^n)*(2^(2*n+1)-1)*Gamma(2*n+1)*Zeta(2*n+1)).
%F a(n) = (-1)^(n+1)*2^(2*n)*i*Li_(2*n,i) where i=sqrt(-1) and Li is polylogarithm function.
%F a(n) = (-64)^n*(zeta(-2*n,1/4)-zeta(-2*n,3/4)) where zeta is Hurwitz zeta function.
%F a(n) = (-16)^n*lerchphi(-1,-2*n,1/2). (End)
%t Union[ Range[0, 26]! CoefficientList[ Series[ Sin[x]^2/Cos[ 2x], {x, 0, 26}], x]] (* _Robert G. Wilson v_, Apr 16 2011 *)
%t Table[(-1)^(n + 1) 2^(2 n) I PolyLog[-2 n, I], {n, 1, 13}] (* _Artur Jasinski_, Mar 21 2022 *)
%o (PARI) {a(n) = local(m); if( n<0, 0, m = 2*n; m! * polcoeff( 1 / (2 - 1 / cos(x + x * O(x^m))^2) - 1, m))} /* _Michael Somos_, Apr 16 2011 */
%o (Sage)
%o @CachedFunction
%o def sp(n,x) :
%o if n == 0 : return 1
%o return -add(2^(n-k)*sp(k,1/2)*binomial(n,k) for k in range(n)[::2])
%o def A000816(n) : return 0 if n == 0 else abs(sp(2*n,x)/2)
%o [A000816(n) for n in (0..13)] # _Peter Luschny_, Jul 30 2012
%Y Cf. A000364, A000819, A000822, A000828, A003707, A009125, A009569.
%K nonn
%O 0,2
%A _N. J. A. Sloane_
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