A012079 - OEIS

%I #14 Jan 23 2019 10:50:07

%S 1,-1,-11,-301,-16631,-1620601,-250557251,-56629836901,

%T -17602836565871,-7193368568377201,-3735581618747946491,

%U -2401310609311293289501,-1871136400199563216523111

%N cos(arcsin(tan(x)))=1-1/2!*x^2-11/4!*x^4-301/6!*x^6-16631/8!*x^8...

%F a(n)=-(2*sum(k=1..2*n, (C(k-1)*sum(j=2*k..2*n, binomial(j-1,2*k-1)*j!*2^(2*n-j-2*k)*(-1)^((n+k)+j)*stirling2(2*n,j))))) with n>0, a(0)=1, C(n)=A000108(n) (Catalan numbers). [_Vladimir Kruchinin_, Oct 08 2012]

%t With[{nn=30},Take[CoefficientList[Series[Cos[ArcSin[Tan[x]]],{x,0,nn}],x] Range[0,nn]!,{1,-1,2}]] (* _Harvey P. Dale_, Jan 23 2019 *)

%o (Maxima) a[n]:=if n=0 then 1 else -2*sum(binomial(2*k-2,k-1)/k*sum(binomial(j-1,2*k-1)*j!*2^(2*n-j+(-2)*k)*(-1)^(n+k+j)*stirling2(2*n,j), j, 2*k, 2*n), k, 1, 2*n); makelist(a[n], n, 0, 12); [_Vladimir Kruchinin_, Oct 08 2012]

%Y Bisection of A012253.

%K sign

%O 0,3

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