A012085 Even coefficients in expansion of e.g.f. cos(x)/sqrt(cos(2*x)).

1, 1, 17, 721, 58337, 7734241, 1526099057, 419784870961, 153563504618177, 72104198836466881, 42270463533824671697, 30262124466958766778001, 25981973075048213029395617, 26350476755161831091778460321

Offset: 0

Author

Patrick Demichel (patrick.demichel(AT)hp.com)

Keywords

• nonn

Examples

			sec(arcsin(tan(x))) = 1 + 1/2!*x^2 + 17/4!*x^4 + 721/6!*x^6 + 58337/8!*x^8...
		

Programs

Mathematica

Table[n!*SeriesCoefficient[Cos[x]/Sqrt[Cos[2*x]],{x,0,n}],{n,0,30,2}] (* Vaclav Kotesovec, Oct 07 2013 *)

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 */

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 */

Formula

E.g.f.: Sum_{k>=0} a(k)x^(2k)/(2k)! = cos(x)/sqrt(cos(2*x)) = sec(arcsin(tan(x))).

a(n) ~ 2*sqrt(2/Pi) * n^(2*n) * (8/Pi)^(2*n) / exp(2*n). - Vaclav Kotesovec, Oct 07 2013

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