A306656 Number of ways to fill a 3D matrix with n distinct values.

1, 1, 6, 18, 144, 360, 6480, 15120, 403200, 2177280, 32659200, 119750400, 8622028800, 18681062400, 784604620800, 11769069312000, 313841848320000, 1067062284288000, 115242726703104000, 364935301226496000, 43792236147179520000, 459818479545384960000

Offset: 0

Author

Yigit Oktar, Mar 03 2019

Keywords

• nonn

Comments

This sequence is a generalization of A323295 to the 3D case. Usually, in multidimensional data related applications (i.e., images, MRI), data is vectorized and then processed. However, because of vectorization, the spatial information in the data is lost. This reverse mapping shows the possible number of spatial states the original data could have been in.

Examples

			For n = 6, a(6) = 6480, A007425(6) = 9 namely there are 9 ways to arrange 6 voxels into a 3D matrix: [1,1,6], [1,6,1], [6,1,1], [2,3,1], [3,2,1], [2,1,3], [3,1,2], [1,2,3], [1,3,2]. Then there are 6! ways to fill it with the numbers. 9*6! = 6480.
		

Crossrefs

Cf. A007425, A077592, A323295.

Programs

Maple

with(numtheory):
a:= n-> `if`(n=0, 1, add(tau(d), d=divisors(n))*n!):
seq(a(n), n=0..23);  # Alois P. Heinz, Mar 03 2019

Mathematica

A007425[n_] := DivisorSigma[0, #]& /@ Divisors[n] // Total;
a[n_] := If[n == 0, 1, A007425[n]*n!];
Table[a[n], {n, 0, 21}] (* Jean-François Alcover, Feb 02 2025 *)

Formula

a(n) = A007425(n) * n! for n > 0, a(0) = 1.

Extensions

More terms from Alois P. Heinz, Mar 03 2019