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

0,4

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

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

Sequence in context: A090594 A361912 A188496 * A191501 A085549 A022492

Adjacent sequences: A369076 A369077 A369078 * A369080 A369081 A369082

Keyword

nonn

Author

Alois P. Heinz, Jan 12 2024

Status

approved