A363550 - OEIS

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

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OFFSET

0,3

COMMENTS

The blocks are ordered with increasing least elements.

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..1153

Wikipedia, Partition of a set

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.

EXAMPLE

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.

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);

CROSSREFS

Column k=1 of A363519.

Cf. A000110, A011968.

Sequence in context: A082833 A101709 A005247 * A355715 A280104 A135259

Adjacent sequences: A363547 A363548 A363549 * A363551 A363552 A363553

KEYWORD

nonn

AUTHOR

Alois P. Heinz, Jun 09 2023

STATUS

approved