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Expansion of e.g.f. arcsin(tan(x)) (odd powers only).
4

%I #15 Sep 02 2022 09:46:02

%S 1,3,45,1743,133305,17089083,3302755365,896199578823,324729845802225,

%T 151401766241310963,88276508686397289885,62925559543228826845503,

%U 53835082550295989275314345,54438337988081689498005862443,64228314189095958231926869651605

%N Expansion of e.g.f. arcsin(tan(x)) (odd powers only).

%C arcsin(sec(x)*sin(x)) = x + 3/3!*x^3 + 45/5!*x^5 + 1743/7!*x^7 + ...

%C arccos(tan(x)) = Pi/2 - x - 3*x^3/3! - 45*x^5/5! - 1743*x^7/7! - ...

%F (8 + z1)*z3 = - 96*z2 + 9*z2^2 - 256*z1 + 72*z2*z1 + 288*z1^2 + 6*z2*z1^2 + 48*z1^2 + z1^4 where z1 = f'(x)^2, z2 = f''(x)^2, z3 = f'''(x)^2, and f(x) = arcsin(tan(x)). - _Michael Somos_, Sep 01 2022

%F a(n) = (2n+1)! * [x^(2n+1)] arcsin(tan(x)). - _Alois P. Heinz_, Sep 02 2022

%t a[ n_] := If[ n<0, 0, (2*n+1)! * SeriesCoefficient[ ArcSin @ Tan @ x, {x, 0, 2*n+1}]]; (* _Michael Somos_, Sep 01 2022 *)

%o (PARI) {a(n) = if( n<0, 0, (2*n+1)! * polcoeff( asin( tan(x + O(x^(2*n+2)))), 2*n+1))}; /* _Michael Somos_, Sep 01 2022 */

%K nonn

%O 0,2

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

%E Name clarified by _Joerg Arndt_, Sep 02 2022