A115415 Real part of (n + i)^n, with i=sqrt(-1).

1, 1, 3, 18, 161, 1900, 27755, 482552, 9722113, 222612624, 5707904499, 161981127968, 5039646554593, 170561613679808, 6237995487261915, 245159013138710400, 10303367499652761601, 461102348510408544512, 21891769059478538933603, 1098983344602124698522112

Offset

0,3

Links

Seiichi Manyama, Table of n, a(n) for n = 0..386

Formula

a(n) = n! * [x^n] exp(n*x)*cos(x). - Ilya Gutkovskiy, Apr 10 2018

a(n) ~ cos(1) * n^n. - Vaclav Kotesovec, Jun 08 2019

a(n) = Sum_{j=0..floor(n/2)} binomial(n,2j)*n^(n-2j)*(-1)^j. - Chai Wah Wu, Feb 15 2024

a(n) = (1/2)*((n + i)^n + (n - i)^n) where i is the imaginary unit. - Gerry Martens, Dec 30 2024

a(n) = (n^2 + 1)^(n/2) * cos(n * arg(n + i)). - Amiram Eldar, Dec 12 2025

Mathematica

Table[ Re[(n + I)^n], {n, 0, 17}] (* Robert G. Wilson v, Jan 23 2006 *)

Prog

(PARI) a(n) = real((n + I)^n); \\ Michel Marcus, Apr 11 2018
(Python)
from math import comb
def A115415(n): return sum(comb(n, j)*n**(n-j)*(-1 if j&2 else 1) for j in range(0, n+1, 2)) # Chai Wah Wu, Feb 15 2024

Crossrefs

Cf. A000312, A009545, A115416 (imaginary part), A121626, A370189.

Sequence in context: A309985 A328030 A301371 * A364432 A065058 A032031

Adjacent sequences: A115412 A115413 A115414 * A115416 A115417 A115418

Keyword

nonn,easy

Author

Reinhard Zumkeller, Jan 22 2006

Extensions

More terms from Robert G. Wilson v, Jan 23 2006

Status

approved