OFFSET
0,2
COMMENTS
arcsin(sec(x)*sin(x)) = x + 3/3!*x^3 + 45/5!*x^5 + 1743/7!*x^7 + ...
arccos(tan(x)) = Pi/2 - x - 3*x^3/3! - 45*x^5/5! - 1743*x^7/7! - ...
FORMULA
(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
a(n) = (2n+1)! * [x^(2n+1)] arcsin(tan(x)). - Alois P. Heinz, Sep 02 2022
MATHEMATICA
a[ n_] := If[ n<0, 0, (2*n+1)! * SeriesCoefficient[ ArcSin @ Tan @ x, {x, 0, 2*n+1}]]; (* Michael Somos, Sep 01 2022 *)
PROG
(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 */
CROSSREFS
KEYWORD
nonn
AUTHOR
Patrick Demichel (patrick.demichel(AT)hp.com)
EXTENSIONS
Name clarified by Joerg Arndt, Sep 02 2022
STATUS
approved