A356568 a(n) = (4^n - 1)*n^(2*n).

0, 3, 240, 45927, 16711680, 9990234375, 8913923665920, 11111328602485167, 18446462598732840960, 39346257980661240576303, 104857500000000000000000000, 341427795961470170556885610263, 1333735697353436921058237339402240, 6156119488473827117528057630000587767

Offset: 0

Author

Enrique Navarrete, Sep 30 2022

Keywords

• nonn
• easy

Comments

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.

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.

Examples

			For n=1, the functions are f1: (1,1),(2,1); f2: (1,2),(2,2); f3: (1,2),(2,1).
		

Links

• Sidney Cadot, Table of n, a(n) for n = 0..30

Crossrefs

Cf. A062206, A085534.

Programs

Mathematica

a[n_] := If[n == 0, 0, (4^n - 1)*n^(2*n)] (* Sidney Cadot, Jan 05 2023 *)
Join[{0},Table[(4^n-1)n^(2n),{n,20}]] (* Harvey P. Dale, Mar 09 2025 *)

PARI

a(n) = (4^n - 1)*n^(2*n) \\ Charles R Greathouse IV, Oct 03 2022

Python

def A356568(n): return ((1<<(m:=n<<1))-1)*n**m # Chai Wah Wu, Nov 18 2022

Formula

a(n) = A085534(n) - A062206(n).