%I #29 Dec 19 2017 18:37:37

%S 1,2,4,16,104,624,3648,29184,302976,3029760,29698560,356382720,

%T 5111976960,71567677440,986336870400,15781389926400,289206418636800,

%U 5205715535462400,92506221468057600,1850124429361152000,41285515024760832000,908281330544738304000

%N Real part of n!*Sum_{k=1..n} i^(k-1)/k, where i is sqrt(-1).

%H Iain Fox, <a href="/A281964/b281964.txt">Table of n, a(n) for n = 1..450</a> (first 100 terms from Daniel Suteu)

%F a(n) ~ Pi/4 * n!.

%F a(1) = 1, a(n+1) = a(n)*(n+1) + n!*cos(Pi*n/2).

%F E.g.f.: arctan(x)/(1 - x). - _Ilya Gutkovskiy_, Dec 19 2017

%e For n=5, a(5) = 104, which is the real part of 5!*(1/1 + i/2 - 1/3 - i/4 + 1/5) = 104+30*i.

%o (PARI) a(n) = real(n!*sum(k=1, n, I^(k-1)/k));

%o (PARI) first(n) = x='x+O('x^(n+1)); Vec(serlaplace(atan(x)/(1 - x))) \\ _Iain Fox_, Dec 19 2017

%Y The corresponding imaginary part is A282132.

%Y Cf. A003881, A024167.

%K nonn

%O 1,2

%A _Daniel Suteu_, Feb 06 2017