A012458 Expansion of e.g.f. arcsin(arctan(x)^2), even powers only.

2, -16, 488, -30336, 3303072, -556926720, 135234491520, -44790227865600, 19434264780034560, -10703464305128755200, 7296385856008015226880, -6031102564833737842851840, 5942962739028147945628999680

Offset: 0

Author

Patrick Demichel (patrick.demichel(AT)hp.com)

Keywords

• sign
• changed

Examples

			2/2!*x^2 - 16/4!*x^4 + 488/6!*x^6 - 30336/8!*x^8...
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..210

Programs

Mathematica

With[{nn=30}, Take[CoefficientList[Series[ArcSin[ArcTan[x]^2], {x,0,nn}], x] Range[0,nn]!, {3,-1,2}]] (* Harvey P. Dale, Dec 13 2011 *)

Formula

a(n) ~ (-1)^n * 2^(2*n + 3/2) * (exp(2) + 1)^(2*n + 5/2) * n^(2*n+1) / (exp(2*n+1) * (exp(2) - 1)^(2*n + 3/2)). - Vaclav Kotesovec, Aug 25 2025

Extensions

Definition corrected by Harvey P. Dale, Dec 13 2011

Name edited by Jason Yuen, Aug 25 2025