A012023 - OEIS

%I #18 Jan 29 2018 02:52:45

%S 1,-1,13,-421,25369,-2449801,346065061,-67243537453,17192488230961,

%T -5593309059948049,2255588021494237501,-1103994926592923677621,

%U 644587811150505183179593,-442516027690815793746696601

%N Expansion of e.g.f. cos(sin(arctan(x))) (even powers).

%F a(n) = (2*n)!*(-1)^n*sum(j=0..n, binomial(n-1,n-j)/(2*j)!). - _Vladimir Kruchinin_, May 19 2011

%F a(n) = -(12*n^2-24*n+13)*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 08 2013

%F a(n) ~ (-1)^n * (2*n)^(2*n-1/3) * exp(3/2*(2*n)^(1/3) - 2*n) / sqrt(3) * (1 - 19/72*2^(2/3)/n^(1/3) + 553/5184*2^(1/3)/n^(2/3)). - _Vaclav Kotesovec_, Nov 08 2013

%e cos(sin(arctan(x))) = 1 - (1/2!)*x^2 + (13/4!)*x^4 - (421/6!)*x^6 + (25369/8!)*x^8 - ...

%t With[{nn=30},Take[CoefficientList[Series[Cos[Sin[ArcTan[x]]],{x,0,nn}],x] Range[0,nn]!,{1,-1,2}]] (* _Harvey P. Dale_, May 07 2012 *)

%o (Maxima)

%o a(n):=(2*n)!*(-1)^n*sum(binomial(n-1,n-j)/(2*j)!,j,0,n); /* _Vladimir Kruchinin_, May 19 2011 */

%K sign

%O 0,3

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