A012780 Expansion of e.g.f. arcsin(tan(x)) (odd powers only).

1, 3, 45, 1743, 133305, 17089083, 3302755365, 896199578823, 324729845802225, 151401766241310963, 88276508686397289885, 62925559543228826845503, 53835082550295989275314345, 54438337988081689498005862443, 64228314189095958231926869651605

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! - ...

Conjecture: the sequence {a(n)} reduced modulo a positive integer k is eventually periodic with the period dividing phi(k). For example, the sequence taken modulo 7 begins [1, 3, 3, 0, 4, 4, 0, 3, 3, 0, 4, 4, 0, ...], with an apparent period of 6 = phi(7) beginning at a(1). - Peter Bala, Feb 15 2026

Links

Table of n, a(n) for n=0..14.

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

Maple

d := proc (n, x) option remember; if n = 0 then arcsin(x) else simplify((1 + x^2)*diff(d(n-1, x), x)) end if end proc:
seq( eval(d(2*n+1, x), x = 0), n = 0..20); # Peter Bala, Feb 17 2026

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

Cf. A003706.

Sequence in context: A144951 A079484 A012494 * A072503 A154242 A331710

Adjacent sequences: A012777 A012778 A012779 * A012781 A012782 A012783

Keyword

nonn

Author

Patrick Demichel

Extensions

Name clarified by Joerg Arndt, Sep 02 2022

Status

approved