A121626 -id:A121626 - cp's OEIS frontend

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

Reinhard Zumkeller, Jan 22 2006

nonn

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

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

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

a(n) = real((n + I)^n); \\ Michel Marcus, Apr 11 2018

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

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

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

A121625 Real part of (n + n*i)^n.

Original entry on oeis.org

1, 1, 0, -54, -1024, -12500, 0, 6588344, 268435456, 6198727824, 0, -9129973459552, -570630428688384, -19384006821904192, 0, 56050417968750000000, 4722366482869645213696, 211773507042902211629312, 0, -1012950863698080557631477248, -107374182400000000000000000000

Offset: 0

Gary W. Adamson, Aug 12 2006

sign

			a(7) = 6588344 since (7 + 7i)^7 = (6588344 - 6588344i).

Cf. A115415, A121626, A146559.

a[n_] := Re[(n + n*I)^n]; Array[a, 19] (* Robert G. Wilson v, Aug 17 2006 *)

a(n) = real((n + n*I)^n); \\ Michel Marcus, Dec 19 2020

def A121625(n): return n**n*((1, 1, 0, -2)[n&3]<<((n>>1)&-2))*(-1 if n&4 else 1) # Chai Wah Wu, Feb 16 2024

a(n) = Re(n + n*i)^n.
From Chai Wah Wu, Feb 15 2024: (Start)
a(n) = n^n*Re((1+i)^n) = n^n*A146559(n) = n^n*Sum_{n=0..floor(n/2)} binomial(n,2j)*(-1)^j.
a(n) = 0 if and only if n==2 mod 4, as (1+i)^2=2i is purely imaginary, (1+i)^4=-4 is a nonzero real and (1+i) and (1+i)^3=-2+2i both have nonzero real parts.
(End)

More terms from Robert G. Wilson v, Aug 17 2006

a(0)=1 prepended by Alois P. Heinz, Dec 19 2020

A370189 Imaginary part of (1 + n*i)^n, where i is the imaginary unit.

Original entry on oeis.org

0, 1, 4, -18, -240, 1900, 42372, -482552, -14970816, 222612624, 8825080100, -161981127968, -7809130867824, 170561613679808, 9678967816041188, -245159013138710400, -16000787866533953280, 461102348510408544512, 34017524842099233036996, -1098983344602124698522112, -90417110945911655996319600

Offset: 0

Hugo Pfoertner, Feb 14 2024

The ratio a(n)/A121626(n) converges to c for odd n and to -1/c for even n for n -> oo with c = 0.6420926... (= cot(1) (A073449) from Moritz Firsching, Feb 14 2024). See linked plots.

Paolo Xausa, Table of n, a(n) for n = 0..350
Hugo Pfoertner, Plot of ratio a(n)/A121626(n), using Plot 2.
Hugo Pfoertner, Plot of asinh(a(n)) vs asinh(A121626(n)), using Plot 2.

Cf. A121626 (real part), A115415, A115416.

Cf. A073449.

Array[Im[(1+#*I)^#] &, 25, 0] (* Paolo Xausa, Feb 19 2024 *)

PARI
```
a370189(n) = imag((1+I*n)^n)
```

from math import comb
def A370189(n): return sum(comb(n,j)*n**j*(-1 if j-1&2 else 1) for j in range(1,n+1,2)) # Chai Wah Wu, Feb 15 2024

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

cp's OEIS Frontend

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

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A121625 Real part of (n + n*i)^n.

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A370189 Imaginary part of (1 + n*i)^n, where i is the imaginary unit.

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