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%I #36 Nov 01 2021 13:45:42
%S 0,2,4,152,8944,933152,151557184,35402298752,11250504212224,
%T 4668840721981952,2451963626804184064,1589715293557268682752,
%U 1247113599659216858312704,1164315843409068590677041152,1275742921918699248939411718144,1621172561651122048792832473137152
%N E.g.f. 1-sqrt(cos(2*x)) (even part).
%F a(n) = sum((k=1..2*n, binomial(2*k-2,k-1)*2^(2*n-2*k+2)*sum(j=1..k, ((sum(i=0..(j-1)/2, (j-2*i)^(2*n)*binomial(j,i)))*binomial(k,j)*(-1)^(n-j))/2^j))/k).
%F a(n) = 2*sum(k=1..2*n, C(k-1)*sum(i=0..k-1, (i-k)^(2*n)*binomial(2*k,i)*(-1)^(n+k-i))*2^(2*n-3*k+1)), where C(k) = A000108(k). - _Vladimir Kruchinin_, Oct 05 2012
%F G.f.: 1 - 1/U(0) where U(k)= 1 - (2*k-1)*(2*k+2)*x/U(k+1); (continued fraction, due to T. J. Stieltjes). - _Sergei N. Gladkovskii_, Nov 09 2012
%F G.f.: T(0)+1, where T(k) = -1 + x*(2*k-1)*(2*k+2)/( x*(2*k-1)*(2*k+2) + 1/T(k+1) ); (continued fraction). - _Sergei N. Gladkovskii_, Oct 25 2013
%F a(n) ~ (2*n)! * 2^(4*n-2)/(n^(3/2)*Pi^(2*n)). - _Vaclav Kotesovec_, Nov 07 2013
%t Table[n!*SeriesCoefficient[1-Sqrt[Cos[2*x]],{x,0,n}],{n,0,40,2}] (* _Vaclav Kotesovec_, Nov 07 2013 *)
%t With[{nn=40},Take[CoefficientList[Series[1-Sqrt[Cos[2x]],{x,0,nn}],x] Range[0,nn]!,{1,-1,2}]] (* _Harvey P. Dale_, Nov 01 2021 *)
%o (Maxima)
%o a(n):=sum((binomial(2*k-2,k-1)*2^(2*n-2*k+2)*sum(((sum((j-2*i)^(2*n) *binomial(j,i),i,0,(j-1)/2))*binomial(k,j)*(-1)^(n-j))/2^j,j,1,k))/k,k,1,2*n);
%K nonn
%O 0,2
%A _Vladimir Kruchinin_, Jun 22 2011
%E Corrected and extended by _Harvey P. Dale_, Nov 01 2021
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