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%I #26 May 13 2017 23:58:22
%S 1,5,121,6845,698161,111973685,25947503401,8200346492525,
%T 3389281372287841,1774459993676715365,1147649139272698443481,
%U 898537335398420151634205,837511978485668107020082321
%N tan(arcsin(tan(x))) = x+5/3!*x^3+121/5!*x^5+6845/7!*x^7+698161/9!*x^9...
%H Robert Israel, <a href="/A012077/b012077.txt">Table of n, a(n) for n = 0..215</a>
%F 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
%F From _Peter Luschny_, May 13 2017 (Start)
%F a(n) = (2*n+1)! [x^(2*n+1)] tan(x)/sqrt(1-tan(x)^2),
%F a(n) = (2*n+1)! [x^(2*n+1)] tan(arcsin(tan(x))),
%F a(n) = (2*n+1)! [x^(2*n+1)] sinh(arctanh(tan(x))).
%F (End)
%p S:= series(tan(x)/sqrt(1-tan(x)^2), x, 102):
%p seq(coeff(S,x,2*j+1)*(2*j+1)!, j=0..50); # _Robert Israel_, May 08 2017
%t 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 *)
%t 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 *)
%o (Maxima)
%o 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 */
%K nonn
%O 0,2
%A Patrick Demichel (patrick.demichel(AT)hp.com)
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