A160564 Sum of products of factorials of parts times the factorial of the number of parts in all integer partitions of n.

1, 1, 4, 16, 80, 420, 2592, 17352, 132240, 1117200, 10559040, 110276352, 1268640000, 15923168640, 216767367936, 3178157607936, 49918919122944, 835744605027840, 14852897362759680, 279172076525153280, 5531978038112409600, 115241366146485749760

Offset: 0

Author

Geoffrey Critzer, May 19 2009

Keywords

• nonn

Comments

Take each Ferrers diagram of the partitions of n, label the cells within each row and then linearly order the rows.

Examples

			a(3) = 16 because the partitions of 3 can be so ordered in 16 ways: 3 (6); 2,1 (4); 1,1,1 (6).
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..450

Crossrefs

Cf. A101880, A077365, A126787.

Programs

Maple

b:= proc(n, i, p) option remember; `if`(n=0, p!, `if`(i<1, 0,
      add(b(n-i*j, i-1, p+j)*i!^j, j=0..n/i)))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..23);  # Alois P. Heinz, Oct 02 2017

Mathematica

p = Table[Map[Function[n, Apply[Times, n! ]], Partitions[i]], {i, 0, 20}]; q = Table[Map[Function[n, Length[n]! ], Partitions[i]], {i, 0, 20}]; Map[Function[n, Apply[Plus, n]], p*q]