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A361084 Number of partitions of [n] such that in each block the smallest element and the largest element have opposite parities.

%I A361084 #17 Jun 12 2023 14:54:21
%S A361084 1,0,1,0,3,2,17,28,171,430,2617,8496,54739,214714,1477153,6743204,
%T A361084 49550011,256645926,2010328585,11602635128,96590823907,612918061426,
%U A361084 5404902119025,37319203169580,347468152001739,2589081441826334,25375080898848729,202668739104752960
%N A361084 Number of partitions of [n] such that in each block the smallest element and the largest element have opposite parities.
%H A361084 Alois P. Heinz, <a href="/A361084/b361084.txt">Table of n, a(n) for n = 0..165</a>
%H A361084 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%F A361084 a(n) mod 2 = A059841(n).
%e A361084 a(0) = 1: () the empty partition.
%e A361084 a(1) = 0.
%e A361084 a(2) = 1: 12.
%e A361084 a(3) = 0.
%e A361084 a(4) = 3: 1234, 12|34, 14|23.
%e A361084 a(5) = 2: 134|25, 14|235.
%e A361084 a(6) = 17: 123456, 1234|56, 1236|45, 124|356, 1256|34, 12|3456, 12|34|56, 12|36|45, 1346|25, 136|245, 1456|23, 146|235, 14|23|56, 16|2345, 16|23|45, 14|25|36, 16|25|34.
%e A361084 a(7) = 28: 12356|47, 1236|457, 12|356|47, 12|36|457, 13456|27, 1346|257, 134|2567, 134|25|67, 134|27|56, 1356|247, 136|2457, 136|25|47, 136|27|45, 1456|237, 146|2357, 14|23567, 14|235|67, 14|237|56, 156|2347, 16|23457, 156|23|47, 16|235|47, 16|237|45, 16|23|457, 14|257|36, 14|27|356, 156|27|34, 16|257|34.
%p A361084 b:= proc(n, x, y, u, v) option remember; `if`(x+v>n, 0, `if`(n=0, 1,
%p A361084       `if`(y=0, 0, b(n-1, v, u, y-1, x+1)*y)+b(n-1, v, u, y, x+1)+
%p A361084       `if`(v=0, 0, b(n-1, v-1, u+1, y, x)*v)+b(n-1, v, u, y, x)*(u+x)))
%p A361084     end:
%p A361084 a:= n-> b(n, 0$4):
%p A361084 seq(a(n), n=0..30);
%Y A361084 Cf. A000110, A059841, A363592.
%K A361084 nonn
%O A361084 0,5
%A A361084 _Alois P. Heinz_, Jun 12 2023