A012019 E.g.f.: exp(sin(arctan(x))).

1, 1, 1, -2, -11, 16, 301, -104, -15287, -20096, 1239481, 4427776, -146243459, -954111872, 23567903269, 243390205696, -4951201340399, -75389245067264, 1307274054385393, 28248828019830784, -420773143716828539

Offset: 0

Author

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

Keywords

• sign

Examples

			exp(sin(arctan(x))) = 1+x+1/2!*x^2-2/3!*x^3-11/4!*x^4+16/5!*x^5+...
		

Programs

Mathematica

CoefficientList[Series[E^(x/Sqrt[1+x^2]), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Nov 08 2013 *)

Maxima

a(n):=(n!*sum((binomial((n-2)/2,(n-k)/2)*(-1)^((n-k)/2)*((-1)^(n-k)+1))/k!,k,1,n))/2; /* Vladimir Kruchinin, May 18 2011 */

Formula

a(n) = (n!*sum(k=1..n, (C((n-2)/2,(n-k)/2)*(-1)^((n-k)/2)*((-1)^(n-k)+1))/k!))/2, n>0, a(0)=1. - Vladimir Kruchinin, May 18 2011

E.g.f.: exp(x/sqrt(1+x^2)). - Vaclav Kotesovec, Nov 08 2013

a(n) = -(3*n^2 - 12*n + 11)*a(n-2) - 3*(n-4)*(n-3)^2*(n-2)*a(n-4) - (n-6)*(n-5)*(n-4)^2*(n-3)*(n-2)*a(n-6). - Vaclav Kotesovec, Nov 09 2013

Lim sup n->infinity |a(n)|/(2*n^(n-1/3)*exp(3/4*n^(1/3)-n)/sqrt(3)) = 1. - Vaclav Kotesovec, Nov 09 2013

Limit n->infinity a(n)/(2*n^(n-1/3)*exp(3/4*n^(1/3)-n)/sqrt(3)) - cos(3/4*sqrt(3)*n^(1/3) + Pi/6 - Pi/2*mod(n,4)) = 0. - Vaclav Kotesovec, Nov 09 2013