A369079 Number of partitions of [n] such that the element sum of each block is odd.

1, 1, 1, 2, 4, 10, 28, 96, 320, 1436, 5556, 28768, 129600, 730864, 3756936, 23286784, 132872192, 910013776, 5679982288, 42235062784, 286769980416, 2281079563104, 16732506817280, 141975748567040, 1115928688967680, 10077454948692288, 84383735744758464

Offset: 0

Author

Alois P. Heinz, Jan 12 2024

Keywords

• nonn

Comments

Number of partitions of [n] such that each block has an odd number of odd elements.

Examples

			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.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..592
• Wikipedia, Partition of a set

Crossrefs

Column k=2 of A375924.

Cf. A000009, A000110, A152140, A290383, A369080, A374692.

Programs

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);