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

%I A370189 #26 Feb 19 2024 10:49:17
%S A370189 0,1,4,-18,-240,1900,42372,-482552,-14970816,222612624,8825080100,
%T A370189 -161981127968,-7809130867824,170561613679808,9678967816041188,
%U A370189 -245159013138710400,-16000787866533953280,461102348510408544512,34017524842099233036996,-1098983344602124698522112,-90417110945911655996319600
%N A370189 Imaginary part of (1 + n*i)^n, where i is the imaginary unit.
%C A370189 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.
%H A370189 Paolo Xausa, <a href="/A370189/b370189.txt">Table of n, a(n) for n = 0..350</a>
%H A370189 Hugo Pfoertner, <a href="https://oeis.org/plot2a?name1=A370189&amp;name2=A121626&amp;tform1=untransformed&amp;tform2=untransformed&amp;shift=0&amp;radiop1=ratio&amp;drawpoints=true">Plot of ratio a(n)/A121626(n)</a>, using Plot 2.
%H A370189 Hugo Pfoertner, <a href="https://oeis.org/plot2a?name1=A121626&amp;name2=A370189&amp;tform1=asinh&amp;tform2=asinh&amp;shift=0&amp;radiop1=xy&amp;drawpoints=true">Plot of asinh(a(n)) vs asinh(A121626(n))</a>, using Plot 2.
%F A370189 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
%t A370189 Array[Im[(1+#*I)^#] &, 25, 0] (* _Paolo Xausa_, Feb 19 2024 *)
%o A370189 (PARI) a370189(n) = imag((1+I*n)^n)
%o A370189 (Python)
%o A370189 from math import comb
%o A370189 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
%Y A370189 Cf. A121626 (real part), A115415, A115416.
%Y A370189 Cf. A073449.
%K A370189 sign,easy
%O A370189 0,3
%A A370189 _Hugo Pfoertner_, Feb 14 2024