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A363550
Number of partitions of [n] having exactly one parity change within the partition.
2
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
0,3
COMMENTS
The blocks are ordered with increasing least elements.
LINKS
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.
Sequence in context: A082833 A101709 A005247 * A355715 A280104 A135259
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Jun 09 2023
STATUS
approved