OFFSET
0,4
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
EXAMPLE
exp(sin(arctan(x))) = 1+x+1/2!*x^2-2/3!*x^3-11/4!*x^4+16/5!*x^5+...
MATHEMATICA
CoefficientList[Series[E^(x/Sqrt[1+x^2]), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Nov 08 2013 *)
PROG
(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]
CROSSREFS
KEYWORD
sign
AUTHOR
Patrick Demichel (patrick.demichel(AT)hp.com)
STATUS
approved