A362495 Total number of blocks containing at least one odd element and at least one even element in all partitions of [n].

0, 0, 1, 3, 13, 54, 262, 1294, 7109, 40367, 248651, 1587414, 10827740, 76494630, 571499993, 4414720825, 35798107309, 299547765240, 2616358573834, 23536296521084, 220030456297349, 2114721297588097, 21046291460160803, 214984439282684504, 2267305399918683232

Offset: 0

Author

Alois P. Heinz, Jun 05 2023

Keywords

• nonn

Examples

			a(3) = 3 = 1 + 1 + 0 + 1 + 0 : 123, 12|3, 13|2, 1|23, 1|2|3.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..575
• Wikipedia, Partition of a set

Crossrefs

Cf. A005493, A124418, A124425, A138378, A363434.

Programs

Maple

b:= proc(n, x, y, m) option remember; `if`(n=0, m,
      `if`(x+m>0, b(n-1, y, x, m)*(x+m), 0)+b(n-1, y, x+1, m)+
      `if`(y>0, b(n-1, y-1, x, m+1)*y, 0))
    end:
a:= n-> b(n, 0$3):
seq(a(n), n=0..25);

Formula

a(n) = Sum_{k=0..floor(n/2)} k * A124418(n,k).

a(n) = A138378(n) - A363434(n) = A005493(n-1) - A363434(n) for n>=1.