A281964 Real part of n!*Sum_{k=1..n} i^(k-1)/k, where i is sqrt(-1).

1, 2, 4, 16, 104, 624, 3648, 29184, 302976, 3029760, 29698560, 356382720, 5111976960, 71567677440, 986336870400, 15781389926400, 289206418636800, 5205715535462400, 92506221468057600, 1850124429361152000, 41285515024760832000, 908281330544738304000

Offset: 1

Author

Daniel Suteu, Feb 06 2017

Keywords

• nonn

Examples

			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.
		

Links

• Iain Fox, Table of n, a(n) for n = 1..450 (first 100 terms from Daniel Suteu)

Crossrefs

The corresponding imaginary part is A282132.

Cf. A003881, A024167.

Programs

PARI

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

PARI

first(n) = x='x+O('x^(n+1)); Vec(serlaplace(atan(x)/(1 - x))) \\ Iain Fox, Dec 19 2017

Formula

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

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

E.g.f.: arctan(x)/(1 - x). - Ilya Gutkovskiy, Dec 19 2017