A363550 Number of partitions of [n] having exactly one parity change within the partition.

0, 0, 2, 1, 3, 2, 7, 5, 20, 15, 67, 52, 255, 203, 1080, 877, 5017, 4140, 25287, 21147, 137122, 115975, 794545, 678570, 4892167, 4213597, 31858034, 27644437, 218543759, 190899322, 1573857867, 1382958545, 11863100692, 10480142147, 93345011951, 82864869804

Offset: 0

Author

Alois P. Heinz, Jun 09 2023

Keywords

• nonn

Comments

The blocks are ordered with increasing least elements.

Examples

			a(2) = 2: 12, 1|2.
a(3) = 1: 13|2.
a(4) = 3: 134|2, 13|24, 13|2|4.
a(5) = 2: 135|24, 135|2|4.
a(6) = 7: 1356|24, 135|246, 135|24|6, 1356|2|4, 135|26|4, 135|2|46, 135|2|4|6.
a(7) = 5: 1357|246, 1357|24|6, 1357|26|4, 1357|2|46, 1357|2|4|6.
a(8) = 20: 13578|246, 1357|2468, 1357|246|8, 13578|24|6, 1357|248|6, 1357|24|68, 1357|24|6|8, 13578|26|4, 1357|268|4, 1357|26|48, 1357|26|4|8, 13578|2|46, 1357|28|46, 1357|2|468, 1357|2|46|8, 13578|2|4|6, 1357|28|4|6, 1357|2|48|6, 1357|2|4|68, 1357|2|4|6|8.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1153
• Wikipedia, Partition of a set

Crossrefs

Column k=1 of A363519.

Cf. A000110, A011968.

Programs

Maple

b:= proc(n) option remember; `if`(n=0, 1,
      add(b(n-j)*binomial(n-1, j-1), j=1..n))
    end:
a:= n-> `if`(n<2, 0, (h-> b(h)+`if`(n::even, b(h-1), 0))(iquo(n, 2))):
seq(a(n), n=0..35);

Formula

a(0) = a(1) = 0, for n>=2: a(n) = A000110((n-1)/2) if n is odd, a(n) = A011968(n/2) if n is even.

a(2*n) = a(2*n-1) + a(2*n+1) for n>=2.