A364267 Total number of blocks in all set partitions of [n] such that each element is contained in a block whose index parity coincides with the parity of the element.

0, 1, 2, 5, 9, 24, 55, 169, 454, 1567, 4823, 18422, 63609, 265929, 1014266, 4595861, 19143089, 93286964, 420483103, 2189786125, 10601936382, 58688597511, 303349005967, 1776842374930, 9754696729753, 60223101819493, 349624680839546, 2267363687696309

Offset: 0

Author

Alois P. Heinz, Jul 16 2023

Keywords

• nonn

Comments

All odd elements are in blocks with an odd index and all even elements are in blocks with an even index. Blocks are ordered with increasing least elements.

Examples

			a(4) = 9 = 2 + 3 + 4: 13|24, 1|24|3, 1|2|3|4.
a(5) = 24 = 2 + 3 + 3 + 3 + 4 + 4 + 5: 135|24, 13|24|5, 15|24|3, 1|24|35, 15|2|3|4, 1|2|35|4, 1|2|3|4|5.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..660
• Wikipedia, Partition of a set

Crossrefs

Cf. A011655, A274537.

Programs

Maple

b:= proc(n, m, t) option remember; `if`(n=0, m, add(
      `if`(irem(j, 2)=t, b(n-1, max(m, j), 1-t), 0), j=1..m+1))
    end:
a:= n-> b(n, 0, 1):
seq(a(n), n=0..30);

Formula

a(n) = Sum_{k=0..n} k * A274537(n,k).

a(n) mod 2 = A011655(n+1) for n>=1.