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A363429 Number of set partitions of [n] such that each block has at most one even element.

%I A363429 #14 Jun 01 2023 16:48:18
%S A363429 1,1,2,5,10,37,77,372,799,4736,10427,73013,163967,1322035,3017562,
%T A363429 27499083,63625324,646147067,1512354975,16926317722,40012800675,
%U A363429 489109544320,1166271373797,15455199988077,37134022033885,530149003318273,1282405154139046,19619325078384593
%N A363429 Number of set partitions of [n] such that each block has at most one even element.
%H A363429 Alois P. Heinz, <a href="/A363429/b363429.txt">Table of n, a(n) for n = 0..772</a>
%H A363429 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%F A363429 a(n) = Sum_{k=0..ceiling(n/2)} floor(n/2)^k * binomial(ceiling(n/2),k) * Bell(ceiling(n/2)-k).
%e A363429 a(0) = 1: () the empty partition.
%e A363429 a(1) = 1: 1.
%e A363429 a(2) = 2: 12, 1|2.
%e A363429 a(3) = 5: 123, 12|3, 13|2, 1|23, 1|2|3.
%e A363429 a(4) = 10: 123|4, 12|34, 12|3|4, 134|2, 13|2|4, 14|23, 1|23|4, 14|2|3, 1|2|34, 1|2|3|4.
%p A363429 b:= proc(n, m) option remember; `if`(n=0, 1,
%p A363429       b(n-1, m+1)+m*b(n-1, m))
%p A363429     end:
%p A363429 a:= n-> (h-> b(n-h, h))(iquo(n, 2)):
%p A363429 seq(a(n), n=0..30);
%Y A363429 Bisection gives: A134980 (even part).
%Y A363429 Cf. A000110, A110132 (exactly one even), A124421 (at least one even), A363430 (at most one odd).
%K A363429 nonn
%O A363429 0,3
%A A363429 _Alois P. Heinz_, Jun 01 2023