A115416 Imaginary part of (n + i)^n, with i=sqrt(-1).
0, 1, 4, 26, 240, 2876, 42372, 740536, 14970816, 343603216, 8825080100, 250756091552, 7809130867824, 264489160965056, 9678967816041188, 380574552503498624, 16000787866533953280, 716309568462681538816
Offset: 0
Keywords
Links
- Seiichi Manyama, Table of n, a(n) for n = 0..386
Programs
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Maple
seq(Im((n+I)^n), n=0..20); # Robert Israel, Dec 30 2024
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Mathematica
Table[Im[(n + I)^n], {n, 0, 17}] (* Robert G. Wilson v, Jan 23 2006 *) -
PARI
a(n) = imag((n + I)^n); \\ Michel Marcus, Apr 11 2018
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Python
from math import comb def A115416(n): return sum(comb(n,j)*n**(n-j)*(-1 if j-1&2 else 1) for j in range(1,n+1,2)) # Chai Wah Wu, Feb 15 2024
Formula
a(n) = n! * [x^n] exp(n*x)*sin(x). - Ilya Gutkovskiy, Apr 10 2018
a(n) ~ sin(1) * n^n. - Vaclav Kotesovec, Jun 08 2019
a(n) = Sum_{j=0..floor((n-1)/2)} binomial(n,2*j+1)*n^(n-2*j-1)*(-1)^j. - Chai Wah Wu, Feb 15 2024
a(n) = -(i/2)*((n + i)^n - (n - i)^n) where i is the imaginary unit. - Gerry Martens, Dec 30 2024
Extensions
More terms from Robert G. Wilson v, Jan 23 2006