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A363592 Number of partitions of [n] such that in each block the smallest element has the same parity as the largest element.

%I A363592 #14 Jun 12 2023 11:35:17
%S A363592 1,1,1,3,6,20,55,223,761,3595,14532,77818,361605,2155525,11274781,
%T A363592 73822175,428004750,3046519516,19348533739,148493347507,1023481273549,
%U A363592 8412534272415,62450994058052,546699337652602,4343869829492281,40308548641909593,340994681344324137
%N A363592 Number of partitions of [n] such that in each block the smallest element has the same parity as the largest element.
%H A363592 Alois P. Heinz, <a href="/A363592/b363592.txt">Table of n, a(n) for n = 0..150</a>
%H A363592 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%F A363592 a(n) mod 2 = A131719(n+1).
%e A363592 a(0) = 1: () the empty partition.
%e A363592 a(1) = 1: 1.
%e A363592 a(2) = 1: 1|2.
%e A363592 a(3) = 3: 123, 13|2, 1|2|3.
%e A363592 a(4) = 6: 123|4, 13|24, 13|2|4, 1|234, 1|24|3, 1|2|3|4.
%e A363592 a(5) = 20: 12345, 1235|4, 123|4|5, 1245|3, 125|3|4, 1345|2, 135|24, 13|24|5, 135|2|4, 13|2|4|5, 15|234, 1|234|5, 145|2|3, 15|24|3, 1|24|35, 1|24|3|5, 1|2|345, 15|2|3|4, 1|2|35|4, 1|2|3|4|5.
%p A363592 b:= proc(n, x, y, u, v) option remember; `if`(y+u>n, 0, `if`(n=0, 1,
%p A363592       `if`(y=0, 0, b(n-1, v, u, y-1, x+1)*y)+b(n-1, v, u, y, x+1)+
%p A363592       `if`(v=0, 0, b(n-1, v-1, u+1, y, x)*v)+b(n-1, v, u, y, x)*(u+x)))
%p A363592     end:
%p A363592 a:= n-> b(n, 0$4):
%p A363592 seq(a(n), n=0..30);
%Y A363592 Cf. A000110, A131719, A361084.
%K A363592 nonn
%O A363592 0,4
%A A363592 _Alois P. Heinz_, Jun 10 2023