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

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

Offset

0,2

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.

Links

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

Formula

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

Example

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

Mathematica

a[n_] := If[n == 0, 0, (4^n - 1)*n^(2*n)] (* Sidney Cadot, Jan 05 2023 *)

Prog

(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

Crossrefs

Cf. A062206, A085534.

Sequence in context: A142730 A264549 A024044 * A069640 A324444 A236249

Adjacent sequences: A356565 A356566 A356567 * A356569 A356570 A356571

Keyword

nonn,easy

Author

Enrique Navarrete, Sep 30 2022

Status

approved