A012077 tan(arcsin(tan(x))) = x+5/3!*x^3+121/5!*x^5+6845/7!*x^7+698161/9!*x^9...

1, 5, 121, 6845, 698161, 111973685, 25947503401, 8200346492525, 3389281372287841, 1774459993676715365, 1147649139272698443481, 898537335398420151634205, 837511978485668107020082321

Offset: 0

Author

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

Keywords

• nonn

Links

• Robert Israel, Table of n, a(n) for n = 0..215

Programs

Maple

S:= series(tan(x)/sqrt(1-tan(x)^2), x, 102):
seq(coeff(S,x,2*j+1)*(2*j+1)!, j=0..50); # Robert Israel, May 08 2017

Mathematica

nn = 20; Table[(CoefficientList[Series[Tan[x]/Sqrt[1 - Tan[x]^2], {x, 0, 2*nn+1}], x] * Range[0, 2*nn+1]!)[[n]], {n, 2, 2*nn, 2}] (* Vaclav Kotesovec, Feb 06 2015 *)
With[{nn=30},Take[CoefficientList[Series[Tan[ArcSin[Tan[x]]],{x,0,nn}],x] Range[0,nn-1]!,{2,-1,2}]] (* Harvey P. Dale, Mar 22 2015 *)

Maxima

a(n):=sum(binomial(2*m,m)*sum(binomial(j+2*m,2*m)*(j+2*m+1)!*2^(2*n-j-4*m)*(-1)^(n+m+j)*stirling2(2*n+1,j+2*m+1),j,0,2*n-2*m),m,0,n)/((2*n+1)!); /* Vladimir Kruchinin, Jun 15 2011 */

Formula

a(n) = Sum_(m=0..n, binomial(2*m,m)*Sum_(j=0..2*n-2*m, binomial(j+2*m,2*m)*(j+2*m+1)!*2^(2*n-j-4*m)*(-1)^(n+m+j)*stirling2(2*n+1,j+2*m+1))) /((2*n+1)!), n>0. - Vladimir Kruchinin, Jun 15 2011

From Peter Luschny, May 13 2017 (Start)

a(n) = (2*n+1)! [x^(2*n+1)] tan(x)/sqrt(1-tan(x)^2),

a(n) = (2*n+1)! [x^(2*n+1)] tan(arcsin(tan(x))),

a(n) = (2*n+1)! [x^(2*n+1)] sinh(arctanh(tan(x))).

(End)