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

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

Offset: 0

Author

Hugo Pfoertner, Feb 14 2024

Keywords

• sign
• easy

Comments

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.

Links

• 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.

Crossrefs

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

Cf. A073449.

Programs

Mathematica

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

PARI

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

Python

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

Formula

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