A013461 Expansion of e.g.f. exp(arctan(x)-arcsinh(x)).

1, 0, 0, -1, 0, 15, 10, -495, -840, 29015, 87750, -2666475, -12310100, 354343275, 2287605450, -64350661375, -549013374000, 15319760415375, 166031512396750, -4630895520073875, -61922192590147500, 1733150999302701875, 27960041702488691250, -786845800090809834375

Offset

0,6

Links

Robert Israel, Table of n, a(n) for n = 0..449

Formula

D-finite with recurrence: (n + 2)*(n - 1)*(n + 1)^2*a(n) - (n + 2)*(1 + 2*n)*a(n + 1) + (n + 2)*(2*n + 3)*a(n + 2) - 2*a(n + 3) + a(n + 4) = 0. - Robert Israel, Feb 20 2026

Example

1 -1/3!*x^3 +15/5!*x^5 +10/6!*x^6 -495/7!*x^7...

Maple

f:= gfun:-rectoproc({(n + 2)*(n - 1)*(n + 1)^2*a(n) - (n + 2)*(1 + 2*n)*a(n + 1) + (n + 2)*(2*n + 3)*a(n + 2) - 2*a(n + 3) + a(n + 4), a(0)=1, a(1)=0, a(2)=0, a(3)=-1}, a(n), remember):
map(f, [$0..30]); # Robert Israel, Feb 20 2026

Mathematica

With[{nn=23}, CoefficientList[Series[Exp[ArcTan[x]-ArcSinh[x]], {x, 0, nn}], x] Range[0, nn]!] (* Harvey P. Dale, Nov 22 2013 *)

Crossrefs

Sequence in context: A321419 A013379 A013449 * A013430 A131082 A296818

Adjacent sequences: A013458 A013459 A013460 * A013462 A013463 A013464

Keyword

sign

Author

Patrick Demichel

Extensions

Definition clarified by Harvey P. Dale, Nov 22 2013

Status

approved