A012091 cos(arcsin(arctan(x)))=1-1/2!*x^2+5/4!*x^4-109/6!*x^6+4521/8!*x^8...

1, -1, 5, -109, 4521, -330681, 36468717, -5721101541, 1205289090513, -328897528901361, 112769846051251797, -47472161654530776285, 24068680135210975714425, -14466929567049590650593705

Offset: 0

Author

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

Keywords

• sign

Crossrefs

Cf. A012254.

Programs

Mathematica

With[{nn=30},Take[CoefficientList[Series[Cos[ArcSin[ArcTan[x]]],{x,0,nn}],x] Range[0,nn]!,{1,-1,2}]] (* Harvey P. Dale, Feb 07 2015 *)

Maxima

a[n]:=if n=0 then 1 else (2*n)!*sum(((2*k)!*binomial(2*k-2,k-1)*(-1)^(n+k+1)*sum((2^(i+1)*stirling1(i,2*k)*binomial(2*n-1,i-1))/i!, i, 2*k, 2*n))/(k*2^(4*k)), k, 1, 2*n); makelist(a[n], n, 0, 13); /* Vladimir Kruchinin, Oct 08 2012 */

Formula

a(n) = (2*n)!*sum(k=1..2*n, ((2*k)!*binomial(2*k-2,k-1)*(-1)^(n+k+1)*sum(i=2*k..2*n, (2^(i+1)*stirling1(i,2*k)*binomial(2*n-1,i-1))/i!))/(k*2^(4*k))) with n>0, a(0)=1. [Vladimir Kruchinin, Oct 08 2012]