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

%I A290384 #28 Sep 28 2021 19:16:22
%S A290384 1,1,1,3,5,23,57,355,1165,9135,37313,352667,1723605,19063207,
%T A290384 108468169,1374019539,8920711325,127336119839,928899673425,
%U A290384 14751357906571,119445766884325,2088674728868631,18588486479073881,354892573941671363,3443175067395538605
%N A290384 Number of ordered set partitions of [n] such that the smallest element of each block is odd.
%C A290384 All terms are odd.
%H A290384 Alois P. Heinz, <a href="/A290384/b290384.txt">Table of n, a(n) for n = 0..494</a>
%H A290384 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%F A290384 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
%F A290384 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
%e A290384 a(3) = 3: 123, 12|3, 3|12.
%e A290384 a(4) = 5: 1234, 124|3, 3|124, 12|34, 34|12.
%p A290384 b:= proc(n, m, t) option remember; `if`(n=0, m!,
%p A290384       add(b(n-1, max(m, j), 1-t), j=1..m+1-t))
%p A290384     end:
%p A290384 a:= n-> b(n, 0$2):
%p A290384 seq(a(n), n=0..30);
%t A290384 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 *)
%o A290384 (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
%o A290384 (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
%Y A290384 Cf. A000246, A000670, A019538, A290383.
%K A290384 nonn
%O A290384 0,4
%A A290384 _Alois P. Heinz_, Jul 28 2017