This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A012085 #20 Oct 02 2023 13:48:30
%S A012085 1,1,17,721,58337,7734241,1526099057,419784870961,153563504618177,
%T A012085 72104198836466881,42270463533824671697,30262124466958766778001,
%U A012085 25981973075048213029395617,26350476755161831091778460321
%N A012085 Even coefficients in expansion of e.g.f. cos(x)/sqrt(cos(2*x)).
%F A012085 E.g.f.: Sum_{k>=0} a(k)x^(2k)/(2k)! = cos(x)/sqrt(cos(2*x)) = sec(arcsin(tan(x))).
%F A012085 a(n) ~ 2*sqrt(2/Pi) * n^(2*n) * (8/Pi)^(2*n) / exp(2*n). - _Vaclav Kotesovec_, Oct 07 2013
%F A012085 a(n) = Sum_{j=0..n} Sum_{k=0..j} (2*n+1)!*(4*k-2*j+1)^(2*n)/(n!*(n-j)!*k!*(j-k)!*(2*j+1)*(-2)^j*(-4)^n). - _Tani Akinari_, Oct 02 2023
%e A012085 sec(arcsin(tan(x))) = 1 + 1/2!*x^2 + 17/4!*x^4 + 721/6!*x^6 + 58337/8!*x^8...
%t A012085 Table[n!*SeriesCoefficient[Cos[x]/Sqrt[Cos[2*x]],{x,0,n}],{n,0,30,2}] (* _Vaclav Kotesovec_, Oct 07 2013 *)
%o A012085 (PARI) {a(n)=local(A); if(n<0, 0, n*=2; A=x*O(x^n); n!*polcoeff( cos(x+A)/sqrt(cos(2*x+A)), n))} /* _Michael Somos_, Jul 18 2005 */
%o A012085 (PARI) {a(n)=sum(j=0,n,sum(k=0,j,(2*n+1)!*(4*k-2*j+1)^(2*n)/(n!*(n-j)!*k!*(j-k)!*(2*j+1)*(-2)^j*(-4)^n)))}; /* _Tani Akinari_, Oct 02 2023 */
%K A012085 nonn
%O A012085 0,3
%A A012085 Patrick Demichel (patrick.demichel(AT)hp.com)