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

0,4

Comments

All terms are odd.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..494

Wikipedia, Partition of a set

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

Example

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

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 *)

Prog

(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

Crossrefs

Cf. A000246, A000670, A019538, A290383.

Sequence in context: A120937 A075307 A100302 * A023247 A027753 A066411

Adjacent sequences: A290381 A290382 A290383 * A290385 A290386 A290387

Keyword

nonn

Author

Alois P. Heinz, Jul 28 2017

Status

approved