A012459 Expansion of e.g.f. arctan(arctan(x)*arctan(x)) (even powers only).

0, 2, -16, 128, 9984, -1444608, 105185280, 11495516160, -7997802086400, 2106956324536320, 3592879276032000, -400446184903952302080, 271017181684548910448640, -56861458491339484126248960

Offset

0,2

Links

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

Formula

a(n) = 4*(2*n)!*sum(m=1..(2*n+1)/2, (2^(-4*m)*(4*m-2)!*(-1)^(n+m)*sum(i=4*m-2..2*n, (2^i*Stirling1(i,4*m-2)*binomial(2*n-1,i-1))/i!))/(2*m-1)). - Vladimir Kruchinin, Jun 13 2011

Example

arctan(arctan(x)*arctan(x)) = (2/2!)*x^2 - (16/4!)*x^4 + (128/6!)*x^6 + (9984/8!)*x^8 - ...

Mathematica

With[{nn=30}, Take[CoefficientList[Series[ArcTan[ArcTan[x]*ArcTan[x]], {x, 0, nn}], x] Range[0, nn]!, {1, -1, 2}]] (* Ray Chandler, Nov 28 2016 *)

Prog

(Maxima)
a(n):=4*(2*n)!*sum((2^(-4*m)*(4*m-2)!*(-1)^(n+m)*sum((2^i*stirling1(i, 4*m-2)*binomial(2*n-1, i-1))/i!, i, 4*m-2, 2*n))/(2*m-1), m, 1, (2*n+1)/2); /* Vladimir Kruchinin, Jun 13 2011 */

Crossrefs

Sequence in context: A333719 A161737 A171451 * A012463 A037517 A037720

Adjacent sequences: A012456 A012457 A012458 * A012460 A012461 A012462

Keyword

sign

Author

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

Status

approved