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%I #16 Aug 30 2025 12:09:53
%S 1,2,11,96,1121,16280,281987,5666304,129488641,3315041568,93958705499,
%T 2920298135040,98749216968481,3608920706225536,141743544911838547,
%U 5953777300691189760,266315973364196014081,12638365012375994704384,634207216217264733599531,33552879853099295377612800
%N a(n) = Sum_{k=0..n} (n-i)^k * (n+i)^(n-k), where i is the imaginary unit.
%H Vincenzo Librandi, <a href="/A387459/b387459.txt">Table of n, a(n) for n = 0..350</a>
%F a(n) = ((1 + i*n)*(-i + n)^n + (1 - i*n)*(i + n)^n)/2, where i is the imaginary unit.
%F For n > 0, a(n) = (1 + n^2)^(n/2) * (cos(n*arctan(1/n)) + n*sin(n*arctan(1/n))).
%F a(n) ~ sin(1) * n^(n+1).
%t Table[Sum[(n-I)^k*(n+I)^(n-k), {k, 0, n}], {n, 0, 20}]
%t (* or *)
%t Table[((1 + I*n)*(-I + n)^n + (1 - I*n)*(I + n)^n)/2, {n, 0, 20}]
%o (PARI) a(n) = sum(k=0, n, (n-I)^k * (n+I)^(n-k)); \\ _Michel Marcus_, Aug 30 2025
%o (Magma) C<I> := ComplexField(); [Floor(Abs( ((1 + I*n)*(-I + n)^n + (1 - I*n)*(I + n)^n)/2)): n in [0..30]]; // _Vincenzo Librandi_, Aug 30 2025
%Y Cf. A387430.
%K nonn
%O 0,2
%A _Vaclav Kotesovec_, Aug 29 2025