A323295 Number of ways to fill a matrix with the first n positive integers.

1, 1, 4, 12, 72, 240, 2880, 10080, 161280, 1088640, 14515200, 79833600, 2874009600, 12454041600, 348713164800, 5230697472000, 104613949440000, 711374856192000, 38414242234368000, 243290200817664000, 14597412049059840000, 204363768686837760000

Offset

0,3

Links

Table of n, a(n) for n=0..21.

Formula

a(n) = A000005(n) * n! for n > 0, a(0) = 1.

E.g.f.: 1 + Sum_{k>=1} x^k/(1 - x^k). - Ilya Gutkovskiy, Sep 13 2019

Example

The a(4) = 72 matrices consist of:
  24 row/column permutations of [1 2 3 4]
+
  4 row/column permutations of [1 2]
                               [3 4]
+
  4 row/column permutations of [1 2]
                               [4 3]
+
  4 row/column permutations of [1 3]
                               [2 4]
+
  4 row/column permutations of [1 3]
                               [4 2]
+
  4 row/column permutations of [1 4]
                               [2 3]
+
  4 row/column permutations of [1 4]
                               [3 2]
+
  24 row/column permutations of [1]
                                [2]
                                [3]
                                [4]

Mathematica

Join[{1}, Table[DivisorSigma[0, n]*n!, {n, 30}]]

Prog

(PARI) a(n) = if (n==0, 1, numdiv(n)*n!); \\ Michel Marcus, Jan 15 2019

Crossrefs

Cf. A000005, A000142, A038041, A053529, A057625, A061095, A120733, A121860.

Cf. A323300, A323301, A323307, A323351.

Sequence in context: A335907 A013195 A356667 * A166746 A052558 A369285

Adjacent sequences: A323292 A323293 A323294 * A323296 A323297 A323298

Keyword

nonn

Author

Gus Wiseman, Jan 12 2019

Status

approved