A363587 Number of partitions of [n] such that in the set of smallest block elements there is an equal number of odd and even terms.

1, 0, 1, 2, 6, 16, 63, 246, 1201, 5632, 30776, 166800, 1032537, 6404960, 44200745, 305485130, 2305218366, 17475547664, 143075155975, 1179769331662, 10409877747841, 92570178170528, 873953428860952, 8318955989166944, 83562716138732321, 846729015766650672

Offset: 0

Author

Alois P. Heinz, Jun 10 2023

Keywords

• nonn

Examples

			a(0) = 1: () the empty partition.
a(1) = 0.
a(2) = 1: 1|2.
a(3) = 2: 13|2, 1|23.
a(4) = 6: 123|4, 134|2, 13|24, 14|23, 1|234, 1|2|3|4.
a(5) = 16: 1235|4, 123|45, 1345|2, 134|25, 135|24, 13|245, 13|2|4|5, 145|23, 14|235, 15|234, 1|2345, 1|23|4|5, 15|2|3|4, 1|25|3|4, 1|2|35|4, 1|2|3|45.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..576
• Wikipedia, Partition of a set

Crossrefs

Cf. A000110, A290383.

Programs

Maple

b:= proc(n, x, y) option remember; `if`(abs(x-y)>2*n, 0,
     `if`(n=0, 1, `if`(y=0, 0, b(n-1, y-1, x+1)*y)+
        b(n-1, y, x)*x + b(n-1, y, x+1)))
    end:
a:= n-> b(n, 0$2):
seq(a(n), n=0..30);

Mathematica

b[n_, x_, y_] := b[n, x, y] = If[Abs[x - y] > 2n, 0, If[n == 0, 1, If[y == 0, 0, b[n-1, y-1, x+1]*y] + b[n-1, y, x]*x + b[n-1, y, x+1]]];
a[n_] := b[n, 0, 0];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Nov 18 2023, after Alois P. Heinz *)