This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A356568 #55 Mar 09 2025 10:30:47
%S A356568 0,3,240,45927,16711680,9990234375,8913923665920,11111328602485167,
%T A356568 18446462598732840960,39346257980661240576303,
%U A356568 104857500000000000000000000,341427795961470170556885610263,1333735697353436921058237339402240,6156119488473827117528057630000587767
%N A356568 a(n) = (4^n - 1)*n^(2*n).
%C A356568 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 A356568 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 A356568 Sidney Cadot, <a href="/A356568/b356568.txt">Table of n, a(n) for n = 0..30</a>
%F A356568 a(n) = A085534(n) - A062206(n).
%e A356568 For n=1, the functions are f1: (1,1),(2,1); f2: (1,2),(2,2); f3: (1,2),(2,1).
%t A356568 a[n_] := If[n == 0, 0, (4^n - 1)*n^(2*n)] (* _Sidney Cadot_, Jan 05 2023 *)
%t A356568 Join[{0},Table[(4^n-1)n^(2n),{n,20}]] (* _Harvey P. Dale_, Mar 09 2025 *)
%o A356568 (PARI) a(n) = (4^n - 1)*n^(2*n) \\ _Charles R Greathouse IV_, Oct 03 2022
%o A356568 (Python)
%o A356568 def A356568(n): return ((1<<(m:=n<<1))-1)*n**m # _Chai Wah Wu_, Nov 18 2022
%Y A356568 Cf. A062206, A085534.
%K A356568 nonn,easy
%O A356568 0,2
%A A356568 _Enrique Navarrete_, Sep 30 2022