A290384 Number of ordered set partitions of [n] such that the smallest element of each block is odd.

1, 1, 1, 3, 5, 23, 57, 355, 1165, 9135, 37313, 352667, 1723605, 19063207, 108468169, 1374019539, 8920711325, 127336119839, 928899673425, 14751357906571, 119445766884325, 2088674728868631, 18588486479073881, 354892573941671363, 3443175067395538605

Offset: 0

Author

Alois P. Heinz, Jul 28 2017

Keywords

• nonn

Comments

All terms are odd.

Examples

			a(3) = 3: 123, 12|3, 3|12.
a(4) = 5: 1234, 124|3, 3|124, 12|34, 34|12.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..494
• Wikipedia, Partition of a set

Crossrefs

Cf. A000246, A000670, A019538, A290383.

Programs

Maple

b:= proc(n, m, t) option remember; `if`(n=0, m!,
      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);

Mathematica

b[n_, m_, t_]:=b[n, m, t]=If[n==0, m!, Sum[b[n - 1, Max[m, j], 1 - t], {j, m + 1 - t}]]; Table[b[n, 0, 0], {n, 0, 50}] (* Indranil Ghosh, Jul 30 2017 *)

PARI

{ A290384(n) = (n==0) + sum(m=0,n, sum(k=1,m+1, stirling(m,k-1,2)*(k-1)! * stirling(n-m,k,2)*k! * (-1)^(m+k+1))); } \\ Max Alekseyev, Sep 28 2021

PARI

{ A290384(n) = polcoef(1 + sum(k=1,n, (-1)^(k-1) / binomial(-1/x-1,k-1) / binomial(1/x-1,k) + O(x^(n+1)) ), n); } \\ Max Alekseyev, Sep 23 2021

Formula

For n>=1, a(n) = Sum_{m=0..n} Sum_{k=1..m+1} (-1)^(m+k+1) * S(m,k-1) * (k-1)! * S(n-m,k) * k! = Sum_{m=0..n} Sum_{k=1..m+1} (-1)^(m+k+1) * A019538(m,k-1) * A019538(n-m,k). - Max Alekseyev, Sep 28 2021

G.f.: 1 + Sum_{k >= 1} (-1)^(k-1) / binomial(-1/x-1,k-1) / binomial(1/x-1,k). - Max Alekseyev, Sep 23 2021