|
%I
%S 1,1,-11,301,-15287,1239481,-146243459,23567903269,-4951201340399,
%T 1307274054385393,-420773143716828539,160635990248839962781,
%U -70764171306270411101351,34822234810202848704345001
%N E.g.f. cosh(sin(arctan(x))) = cosh(x/sqrt(1+x^2)) (even powers only).
%F a(n) = ((2*n)!*sum(k=0..n, C(n-1,n-k)/(2*k)!*(-1)^(n-k))). - _Vladimir Kruchinin_, Jun 17 2011
%F E.g.f.: cosh(x/sqrt(1+x^2)) = 1 + x^2/(G(0) - x^2) where G(k)= (2*k+2)*(2*k+1)*(1+x^2) + x^2 - (2*k+2)*(2*k+1)*x^2*(1+x^2)/G(k+1); (continued fraction, Euler's kind, 1-step). - _Sergei N. Gladkovskii_, Aug 06 2012
%F D-finite with recurrence: a(n) = -(12*n^2 - 24*n + 11)*a(n-1) - 12*(n-2)*(n-1)*(2*n-3)^2*a(n-2) - 16*(n-3)*(n-2)^2*(n-1)*(2*n-5)*(2*n-3)*a(n-3). - _Vaclav Kotesovec_, Nov 09 2013
%F Lim sup n->infinity |a(n)|/(2^(2*n+2/3) * exp(3/4*(2*n)^(1/3)-2*n) * n^(2*n-1/3) / sqrt(3)) = 1. - _Vaclav Kotesovec_, Nov 09 2013
%e cosh(sin(arctan(x))) = 1+1/2!*x^2-11/4!*x^4+301/6!*x^6-15287/8!*x^8...
%t Table[n!*SeriesCoefficient[Cosh[x/Sqrt[1+x^2]],{x,0,n}],{n,0,40,2}] (* _Vaclav Kotesovec_, Nov 08 2013 *)
%o (Maxima)
%o a(n):=((2*n)!*sum(binomial(n-1,n-k)/(2*k)!*(-1)^(n-k),k,0,n)); [_Vladimir Kruchinin_, Jun 17 2011]
%K sign
%O 0,3
%A Patrick Demichel (patrick.demichel(AT)hp.com)
|
