A012097 Even coefficients in expansion of e.g.f. sec(arcsin(arctan(x))).

1, 1, 1, 49, -447, 78561, -5655231, 1014069393, -187639949439, 51144433721793, -16631972691697791, 6837287119716744945, -3357388632038371909695, 1969419543279480607781025

Offset

0,4

Links

Table of n, a(n) for n=0..13.

Peter Bala, The method of Graves for computing inverse functions.

Formula

E.g.f.: A(x) = 1/sqrt(1 - arctan(x)^2). 1/A(x) is the e.g.f. of A012091. - Peter Bala, Jan 28 2026

Example

sec(arcsin(arctan(x))) = 1 + 1/2!*x^2 + 1/4!*x^4+49/6!*x^6 - 447/8!*x^8 + ...

Maple

d := proc (n, x) option remember; if n = 0 then 1/sqrt(1 - x^2) else simplify( (cos(x))^2 * (diff(d(n-1, x), x))) end if end proc:
seq(eval(d(2*n, x), x = 0), n = 0..20); # Peter Bala, Jan 28 2026

Mathematica

With[{nn=30}, Take[CoefficientList[Series[Sec[ArcSin[ArcTan[x]]], {x, 0, nn}], x] Range[0, nn]!, {1, -1, 2}]] (* Harvey P. Dale, Nov 28 2019 *)
nmax = 20; CoefficientList[Series[1/Sqrt[1 - ArcTan[Sqrt[x]]^2], {x, 0, nmax}], x] * (2*Range[0, nmax])! (* Vaclav Kotesovec, Jan 29 2026 *)

Crossrefs

Cf. A012085, A012091.

Sequence in context: A063132 A250081 A250082 * A218594 A263706 A263707

Adjacent sequences: A012094 A012095 A012096 * A012098 A012099 A012100

Keyword

sign,easy

Author

Patrick Demichel

Status

approved