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A370189
Imaginary part of (1 + n*i)^n, where i is the imaginary unit.
2
0, 1, 4, -18, -240, 1900, 42372, -482552, -14970816, 222612624, 8825080100, -161981127968, -7809130867824, 170561613679808, 9678967816041188, -245159013138710400, -16000787866533953280, 461102348510408544512, 34017524842099233036996, -1098983344602124698522112, -90417110945911655996319600
OFFSET
0,3
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
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.
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
MATHEMATICA
Array[Im[(1+#*I)^#] &, 25, 0] (* Paolo Xausa, Feb 19 2024 *)
PROG
(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
CROSSREFS
Cf. A121626 (real part), A115415, A115416.
Cf. A073449.
Sequence in context: A145660 A246531 A278565 * A360283 A214168 A214189
KEYWORD
sign,easy
AUTHOR
Hugo Pfoertner, Feb 14 2024
STATUS
approved