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

%I A290383 #19 Jun 10 2023 19:10:43
%S A290383 1,1,1,2,3,8,17,56,151,584,1893,8360,31499,155720,666169,3633704,
%T A290383 17351967,103284296,543441005,3499082408,20079329875,138860069192,
%U A290383 861908850561,6364334129192,42439075349543,332934707138888,2371469004695797,19681714722718376
%N A290383 Number of set partitions of [n] such that the smallest element of each block is odd.
%C A290383 a(n) + n is odd for all n > 1.
%H A290383 Alois P. Heinz, <a href="/A290383/b290383.txt">Table of n, a(n) for n = 0..607</a>
%H A290383 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%e A290383 a(3) = 2: 123, 12|3.
%e A290383 a(4) = 3: 1234, 124|3, 12|34.
%e A290383 a(5) = 8: 12345, 1234|5, 1245|3, 124|35, 124|3|5, 125|34, 12|345, 12|34|5.
%e A290383 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.
%p A290383 b:= proc(n, m, t) option remember; `if`(n=0, 1,
%p A290383       add(b(n-1, max(m, j), 1-t), j=1..m+1-t))
%p A290383     end:
%p A290383 a:= n-> b(n, 0$2):
%p A290383 seq(a(n), n=0..30);
%p A290383 # second Maple program:
%p A290383 b:= proc(n, m, t) option remember; `if`(n=0, 1,
%p A290383      `if`(t=0, b(n-1, m+1, 1-t), 0)+m*b(n-1, m, 1-t))
%p A290383     end:
%p A290383 a:= n-> b(n, 0$2):
%p A290383 seq(a(n), n=0..30);  # _Alois P. Heinz_, Jan 06 2022
%t A290383 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 *)
%Y A290383 Cf. A000110, A000246, A290384.
%Y A290383 Bisections give: A307375 (even part), A363589 (odd part).
%K A290383 nonn
%O A290383 0,4
%A A290383 _Alois P. Heinz_, Jul 28 2017