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%I #53 Jan 05 2023 18:45:02
%S 0,3,240,45927,16711680,9990234375,8913923665920,11111328602485167,
%T 18446462598732840960,39346257980661240576303,
%U 104857500000000000000000000,341427795961470170556885610263,1333735697353436921058237339402240,6156119488473827117528057630000587767
%N a(n) = (4^n - 1)*n^(2*n).
%C If S = {1,2,3,...,2n}, a(n) is the number of functions from S to S such that at least one even number is mapped to an odd number or at least one odd number is mapped to an even number.
%C Note the result can be obtained as (2*n)^(2*n) - n^(2*n), which is the number of functions from S to S minus the number of functions from S to S that map each even number to an even number and each odd number to an odd number. Hence in particular a(0) = 1-1 = 0.
%H Sidney Cadot, <a href="/A356568/b356568.txt">Table of n, a(n) for n = 0..30</a>
%F a(n) = A085534(n) - A062206(n).
%e For n=1, the functions are f1: (1,1),(2,1); f2: (1,2),(2,2); f3: (1,2),(2,1).
%t a[n_] := If[n == 0, 0, (4^n - 1)*n^(2*n)] (* _Sidney Cadot_, Jan 05 2023 *)
%o (PARI) a(n) = (4^n - 1)*n^(2*n) \\ _Charles R Greathouse IV_, Oct 03 2022
%o (Python)
%o def A356568(n): return ((1<<(m:=n<<1))-1)*n**m # _Chai Wah Wu_, Nov 18 2022
%Y Cf. A062206, A085534.
%K nonn,easy
%O 0,2
%A _Enrique Navarrete_, Sep 30 2022
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