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

0,4

Links

Alois P. Heinz, Table of n, a(n) for n = 0..576

Wikipedia, Partition of a set

Example

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.

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 *)

Crossrefs

Cf. A000110, A290383.

Sequence in context: A150031 A121753 A173994 * A150032 A373452 A283420

Adjacent sequences: A363584 A363585 A363586 * A363588 A363589 A363590

Keyword

nonn

Author

Alois P. Heinz, Jun 10 2023

Status

approved