A279927 Expansion of e.g.f. arctan(x)*exp(x).

0, 1, 2, 1, -4, 9, 110, -279, -4520, 17265, 322618, -1638031, -35226860, 223578809, 5463436134, -41639195623, -1142009233872, 10162622387809, 309463272791538, -3149754003442847, -105510576441518164, 1208991988527548137, 44200537412519181278, -563099647603189449783

Offset: 0

Author

Stanislav Sykora, Jan 06 2017

Keywords

• sign

Examples

			atan(x)*exp(x) = x + 2*x^2/2! + x^3/3! - 4*x^4/4! + 9*x^5/5! + ...
		

Links

• Stanislav Sykora, Table of n, a(n) for n = 0..199

Crossrefs

E.g.f. of exp(x) A000012, -arctan(x) A186246.

Programs

Mathematica

CoefficientList[Series[Exp[x] ArcTan[x], {x,0,12}],x] Range[0,12]!
Table[Sum[Binomial[n+1,2k+1] (-1)^k (n-2k)/(n+1) (2k)!, {k,0,n/2}], {n,0,12}] (* Emanuele Munarini, Dec 16 2017 *)

Maxima

makelist(sum((-1)^k*binomial(n+1,2*k+1)*(n-2*k)/(n+1)*(2*k)!,k,0,floor(n/2)),n,0,12); /* Emanuele Munarini, Dec 16 2017 */

PARI

x='x+O('x^33); concat([0], Vec(serlaplace(atan(x)*exp(x) ) ) ) \\ Joerg Arndt, Jan 06 2017

Formula

From Emanuele Munarini, Dec 16 2017: (Start)

a(n) = Sum_{k=0..n/2} binomial(n+1,2*k+1)*(-1)^k*((n-2*k)/(n+1))*(2k)!.

a(n+3) - a(n+2) + (n+1)*(n+2)*a(n+1) - (n+1)*(n+2)*a(n) = 1.

a(n+4) - 2*a(n+3) + (n^2+5*n+7)*a(n+2) - 2*(n+2)^2*a(n+1) + (n+1)*(n+2)*a(n) = 0. (End)

a(n) ~ (n-1)! * sin(Pi*n/2-1). - Vaclav Kotesovec, Dec 17 2017