A290383 Number of set partitions of [n] such that the smallest element of each block is odd.

1, 1, 1, 2, 3, 8, 17, 56, 151, 584, 1893, 8360, 31499, 155720, 666169, 3633704, 17351967, 103284296, 543441005, 3499082408, 20079329875, 138860069192, 861908850561, 6364334129192, 42439075349543, 332934707138888, 2371469004695797, 19681714722718376

Offset: 0

Author

Alois P. Heinz, Jul 28 2017

Keywords

• nonn

Comments

a(n) + n is odd for all n > 1.

Examples

			a(3) = 2: 123, 12|3.
a(4) = 3: 1234, 124|3, 12|34.
a(5) = 8: 12345, 1234|5, 1245|3, 124|35, 124|3|5, 125|34, 12|345, 12|34|5.
a(6) = 17: 123456, 12346|5, 1234|56, 12456|3, 1245|36, 1246|35, 124|356, 1246|3|5, 124|36|5, 124|3|56, 1256|34, 125|346, 126|345, 12|3456, 126|34|5, 12|346|5, 12|34|56.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..607
• Wikipedia, Partition of a set

Crossrefs

Cf. A000110, A000246, A290384.

Bisections give: A307375 (even part), A363589 (odd part).

Programs

Maple

b:= proc(n, m, t) option remember; `if`(n=0, 1,
      add(b(n-1, max(m, j), 1-t), j=1..m+1-t))
    end:
a:= n-> b(n, 0$2):
seq(a(n), n=0..30);
# second Maple program:
b:= proc(n, m, t) option remember; `if`(n=0, 1,
     `if`(t=0, b(n-1, m+1, 1-t), 0)+m*b(n-1, m, 1-t))
    end:
a:= n-> b(n, 0$2):
seq(a(n), n=0..30);  # Alois P. Heinz, Jan 06 2022

Mathematica

b[n_, m_, t_]:=b[n, m, t]=If[n==0, 1, Sum[b[n - 1, Max[m, j], 1 - t], {j, m + 1 - t}]]; Table[b[n, 0, 0], {n, 0, 50}] (* Indranil Ghosh, Jul 29 2017, after Maple code *)