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

0,3

Links

Alois P. Heinz, Table of n, a(n) for n = 0..250

Wikipedia, Partition of a set

Example

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.

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 *)

Crossrefs

Cf. A000110, A363434, A363435, A363454.

Sequence in context: A180753 A220971 A368182 * A369076 A308519 A205390

Adjacent sequences: A363448 A363449 A363450 * A363452 A363453 A363454

Keyword

nonn

Author

Alois P. Heinz, Jun 02 2023

Status

approved