OFFSET
0,4
COMMENTS
Number of partitions of [n] such that each block has an odd number of odd elements.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..592
Wikipedia, Partition of a set
EXAMPLE
a(0) = 1: the empty partition.
a(1) = 1: 1.
a(2) = 1: 12.
a(3) = 2: 12|3, 1|23.
a(4) = 4: 124|3, 12|34, 14|23, 1|234.
a(5) = 10: 12345, 124|3|5, 12|34|5, 12|3|45, 14|23|5, 1|234|5, 1|23|45, 14|25|3, 1|245|3, 1|25|34.
MAPLE
b:= proc(n, x, y) option remember; `if`(n=0, `if`(y=0, 1, 0),
`if`(n::odd, b(n-1, x+1, y)+`if`(x>0, x*b(n-1, x-1, y+1), 0)+
`if`(y>0, y*b(n-1, x+1, y-1), 0), b(n-1, x, y+1)+(x+y)*b(n-1, x, y)))
end:
a:= n-> b(n, 0$2):
seq(a(n), n=0..26);
# second Maple program:
b:= proc(x, y) option remember; `if`(x+y=0, 1,
add(`if`(j::odd, binomial(x-1, j-1)*add(
b(x-j, y-i)*binomial(y, i), i=0..y), 0), j=1..x))
end:
a:= n-> (h-> b(n-h, h))(iquo(n, 2)):
seq(a(n), n=0..26);
CROSSREFS
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Jan 12 2024
STATUS
approved