A363451 Number of partitions of [n] such that the number of blocks containing only odd elements equals the number of blocks containing only even elements.

1, 0, 2, 2, 9, 23, 99, 353, 1778, 7927, 45273, 238797, 1526331, 9215950, 65020448, 439742641, 3388075807, 25270974635, 210763775071, 1713657668021, 15359474721088, 134902169999841, 1291589459223627, 12165062702520422, 123780591852786693, 1242763745129587332

Offset: 0

Author

Alois P. Heinz, Jun 02 2023

Keywords

• nonn

Examples

			a(0) = 1: () the empty partition.
a(1) = 0.
a(2) = 2: 12, 1|2.
a(3) = 2: 123, 13|2.
a(4) = 9: 1234, 12|34, 12|3|4, 13|24, 14|23, 1|23|4, 14|2|3, 1|2|34, 1|2|3|4.
a(5) = 23: 12345, 123|45, 123|4|5, 125|34, 12|345, 125|3|4, 12|35|4, 134|25, 134|2|5, 135|24, 13|25|4, 13|2|45, 13|2|4|5, 145|23, 14|235, 15|23|4, 1|235|4, 145|2|3, 14|2|35, 15|2|34, 1|2|345, 15|2|3|4, 1|2|35|4.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..250
• Wikipedia, Partition of a set

Crossrefs

Cf. A000110, A363434, A363435, A363454.

Programs

Maple

b:= proc(n, x, y, m) option remember; `if`(n=0, `if`(x=y, 1, 0),
      `if`(x+m>0, b(n-1, y, x, m)*(x+m), 0)+b(n-1, y, x+1, m)+
      `if`(y>0, b(n-1, y-1, x, m+1)*y, 0))
    end:
a:= n-> b(n, 0$3):
seq(a(n), n=0..28);

Mathematica

b[n_, x_, y_, m_] := b[n, x, y, m] = If[n == 0, If[x == y, 1, 0], If[x + m > 0, b[n - 1, y, x, m]*(x + m), 0] + b[n - 1, y, x + 1, m] + If[y > 0, b[n - 1, y - 1, x, m + 1]*y, 0]];
a[n_] := b[n, 0, 0, 0];
Table[a[n], {n, 0, 28}] (* Jean-François Alcover, Oct 20 2023, after Alois P. Heinz *)