A012019 - OEIS

%I #17 Nov 09 2013 14:33:54

%S 1,1,1,-2,-11,16,301,-104,-15287,-20096,1239481,4427776,-146243459,

%T -954111872,23567903269,243390205696,-4951201340399,-75389245067264,

%U 1307274054385393,28248828019830784,-420773143716828539

%N E.g.f.: exp(sin(arctan(x))).

%F 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

%F E.g.f.: exp(x/sqrt(1+x^2)). - _Vaclav Kotesovec_, Nov 08 2013

%F 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

%F 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

%F 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

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

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

%o (Maxima)

%o 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]

%K sign

%O 0,4

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