A013465 Expansion of e.g.f. cos(arctan(x) - arctanh(x)) (even powers only).

1, 0, 0, -160, 0, -691200, 3942400, -14125363200, 295206912000, -902471385088000, 43911376404480000, -140627264197754880000, 12856220870209699840000, -45500991975079280640000000, 6879896103927262740480000000, -27326595042383847793623040000000, 6277006865173170789915033600000000

Offset

0,4

Links

Robert Israel, Table of n, a(n) for n = 0..224

Formula

D-finite with recurrence: 16*n*(2*n + 7)*(2*n + 3)*(n + 3)*(n + 2)*(n + 1)*(1 + 2*n)^2*a(n) + 16*(2*n + 3)*(n + 3)*(n + 2)*(2*n + 7)*a(n + 1) - 8*(2*n + 3)*(n + 3)*(n + 2)*(2*n + 7)*a(n + 2) + a(n + 4) = 0. - Robert Israel, Feb 22 2026

Example

cos(arctan(x) - arctanh(x)) = 1 - 160/6!*x^6 - 691200/10!*x^10 + 3942400/12!*x^12 + ...

Maple

f:= gfun:-rectoproc({16*n*(2*n + 7)*(2*n + 3)*(n + 3)*(n + 2)*(n + 1)*(1 + 2*n)^2*a(n) + 16*(2*n + 3)*(n + 3)*(n + 2)*(2*n + 7)*a(n + 1) - 8*(2*n + 3)*(n + 3)*(n + 2)*(2*n + 7)*a(n + 2) + a(n + 4), a(0)=1, a(1)=0, a(2)=0, a(3)= -160}, a(n), remember):
map(f, [$0..20]); # Robert Israel, Feb 22 2026

Mathematica

With[{nn=30}, Take[CoefficientList[Series[Cos[ArcTan[x]-ArcTanh[x]], {x, 0, nn}], x] Range[0, nn]!, {1, -1, 2}]] (* Ray Chandler, Nov 29 2016 *)

Crossrefs

Sequence in context: A224430 A013454 A013455 * A013466 A376899 A060675

Adjacent sequences: A013462 A013463 A013464 * A013466 A013467 A013468

Keyword

sign

Author

Patrick Demichel

Status

approved