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Imaginary part of (n + n*i)^n where i = sqrt(-1).
1

%I #6 Feb 16 2024 15:21:41

%S 0,1,8,54,0,-12500,-373248,-6588344,0,6198727824,320000000000,

%T 9129973459552,0,-19384006821904192,-1422336873671426048,

%U -56050417968750000000,0,211773507042902211629312,20145360934551827238617088,1012950863698080557631477248,0,-5982809106827246101894271407104

%N Imaginary part of (n + n*i)^n where i = sqrt(-1).

%F a(n) = n^n*A009545(n) = n^n*Sum_{j=0..floor((n-1)/2)} binomial(n,2*j+1)*(-1)^j.

%F a(n) = 0 if and only if n == 0 mod 4.

%F a(4n) = 0.

%F a(4n+1) = (4n+1)^(4n+1)*(-4)^n.

%F a(4n+2) = 2*(4n+2)^(4n+2)*(-4)^n.

%F a(4n+3) = 2*(4n+3)^(4n+3)*(-4)^n.

%o (Python)

%o def A370359(n): return n**n*((0, 1, 2, 2)[n&3]<<((n>>1)&-2))*(-1 if n&4 else 1)

%o (PARI) a(n) = imag((n + n*I)^n); \\ _Michel Marcus_, Feb 16 2024

%Y Cf. A009545, A121625, A146559.

%K sign

%O 0,3

%A _Chai Wah Wu_, Feb 16 2024