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Expansion of e.g.f. arctan(x)*exp(x).
4

%I #26 Mar 08 2024 11:59:47

%S 0,1,2,1,-4,9,110,-279,-4520,17265,322618,-1638031,-35226860,

%T 223578809,5463436134,-41639195623,-1142009233872,10162622387809,

%U 309463272791538,-3149754003442847,-105510576441518164,1208991988527548137,44200537412519181278,-563099647603189449783

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

%H Stanislav Sykora, <a href="/A279927/b279927.txt">Table of n, a(n) for n = 0..199</a>

%F From _Emanuele Munarini_, Dec 16 2017: (Start)

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

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

%F 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)

%F a(n) ~ (n-1)! * sin(Pi*n/2-1). - _Vaclav Kotesovec_, Dec 17 2017

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

%t CoefficientList[Series[Exp[x] ArcTan[x], {x,0,12}],x] Range[0,12]!

%t 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 *)

%o (PARI) x='x+O('x^33); concat([0], Vec(serlaplace(atan(x)*exp(x) ) ) ) \\ _Joerg Arndt_, Jan 06 2017

%o (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 */

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

%K sign

%O 0,3

%A _Stanislav Sykora_, Jan 06 2017