A363472 Total number of blocks in all partitions of [n] where each block has at least one odd element and at least one even element.

0, 0, 1, 1, 5, 13, 55, 193, 941, 4081, 22351, 113761, 694565, 4030153, 27107095, 175738753, 1289775821, 9209233921, 73147903471, 568928274961, 4857161139365, 40796613003433, 372190216061335, 3352314486348433, 32518958606637101, 312271731474218881

Offset: 0

Author

Alois P. Heinz, Jun 05 2023

Keywords

• nonn

Comments

All positive terms are odd.

Examples

			a(2) = 1: 12.
a(3) = 1: 123.
a(4) = 5 = 1 + 2 + 2: 1234, 12|34, 14|23.
a(5) = 13 = 1 + 2 + 2 + 2 + 2 + 2 + 2: 12345, 123|45, 125|34, 12|345, 134|25, 145|23, 14|235.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..580
• Wikipedia, Partition of a set

Crossrefs

Cf. A124425, A362495, A363454.

Programs

Maple

a:= n-> (h-> add(k*Stirling2(h, k)*Stirling2(n-h, k)*k!, k=0..h))(floor(n/2)):
seq(a(n), n=0..30);

Formula

a(n) = Sum_{k=0..floor(n/2)} k * Stirling2(floor(n/2),k) * Stirling2(ceiling(n/2),k) * k!.