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A363430 Number of set partitions of [n] such that each block has at most one odd element.

%I A363430 #13 Jun 01 2023 16:59:53
%S A363430 1,1,2,3,10,17,77,141,799,1540,10427,20878,163967,338233,3017562,
%T A363430 6376149,63625324,137144475,1512354975,3315122947,40012800675,
%U A363430 88981537570,1166271373797,2626214876310,37134022033885,84540738911653,1282405154139046,2948058074576995
%N A363430 Number of set partitions of [n] such that each block has at most one odd element.
%H A363430 Alois P. Heinz, <a href="/A363430/b363430.txt">Table of n, a(n) for n = 0..773</a>
%H A363430 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%F A363430 a(n) = Sum_{k=0..floor(n/2)} ceiling(n/2)^k * binomial(floor(n/2),k) * Bell(floor(n/2)-k).
%e A363430 a(0) = 1: () the empty partition.
%e A363430 a(1) = 1: 1.
%e A363430 a(2) = 2: 12, 1|2.
%e A363430 a(3) = 3: 12|3, 1|23, 1|2|3.
%e A363430 a(4) = 10: 124|3, 12|34, 12|3|4, 14|23, 1|234, 1|23|4, 14|2|3, 1|24|3, 1|2|34, 1|2|3|4.
%e A363430 a(5) = 17: 124|3|5, 12|34|5, 12|3|45, 12|3|4|5, 14|23|5, 1|234|5, 1|23|45, 1|23|4|5, 14|25|3, 14|2|3|5, 1|245|3, 1|24|3|5, 1|25|34, 1|2|34|5, 1|25|3|4, 1|2|3|45, 1|2|3|4|5.
%p A363430 b:= proc(n, m) option remember; `if`(n=0, 1,
%p A363430       b(n-1, m+1)+m*b(n-1, m))
%p A363430     end:
%p A363430 a:= n-> (h-> b(h, n-h))(iquo(n, 2)):
%p A363430 seq(a(n), n=0..30);
%Y A363430 Bisection gives: A134980 (even part).
%Y A363430 Cf. A000110, A110138 (exactly one odd), A124423 (at least one odd), A363429 (at most one even).
%K A363430 nonn
%O A363430 0,3
%A A363430 _Alois P. Heinz_, Jun 01 2023