A282132 -id:A282132

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A281964

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

+10
5

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

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

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

EXAMPLE

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.

PROG

(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

CROSSREFS

The corresponding imaginary part is A282132.

Cf. A003881, A024167.

KEYWORD

nonn

AUTHOR

Daniel Suteu, Feb 06 2017

STATUS

approved

A282567

Imaginary part of A000178(n) * Sum_{k=0..n} i^k/k!, where i = sqrt(-1).

+10
2

0, 1, 2, 10, 240, 29088, 20943360, 105529651200, 4254955536384000, 1544043321627770880000, 5603024405522854969344000000, 223654797931768113135574056960000000, 107131006056993617020920990202331136000000000, 667107003169139201955908457896071963607040000000000000

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

LINKS

Daniel Suteu, Table of n, a(n) for n = 0..50

FORMULA

a(n) ~ sin(1) * A000178(n).

a(0) = 0, a(n) = n!*a(n-1) + A000178(n-1)*sin(Pi/2*n).

Lim_{n->infinity} a(n)/G(n+2) = sin(1), where G(z) is the Barnes G-function.

EXAMPLE

For n = 4, a(4) = 240, which is the imaginary part of A000178(4)*(1/0! + i/1! - 1/2! - i/3! + 1/4!) = 156+240*i.

PROG

(PARI) a(n) = imag(prod(k=0, n, k!) * sum(k=0, n, I^k/k!));

CROSSREFS

The corresponding real part is A282564.

Cf. A000178, A282132.