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A361804 Number of partitions of [n] with an equal number of even and odd block sizes.

%I A361804 #19 Jun 12 2023 18:53:06
%S A361804 1,0,0,3,0,15,45,63,1260,1515,25515,104973,510345,5679765,17252235,
%T A361804 263214318,1207222380,11863296915,101718989235,630468648873,
%U A361804 8281982665215,48583038314415,656006633919945,5122900223419938,54304561161840825,605082149235374265
%N A361804 Number of partitions of [n] with an equal number of even and odd block sizes.
%C A361804 Half the number of block sizes are even and the other half are odd.
%H A361804 Alois P. Heinz, <a href="/A361804/b361804.txt">Table of n, a(n) for n = 0..576</a>
%H A361804 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%F A361804 a(n) mod 5 = 3 for n in { A004767 }, a(n) mod 5 = 1 for n = 0 and a(n) mod 5 = 0 for all other n (n in { A004773 } \ { 0 }).
%F A361804 a(n) mod 3 = 0 for n >= 1.
%e A361804 a(0) = 1: () the empty partition.
%e A361804 a(1) = 0.
%e A361804 a(2) = 0.
%e A361804 a(3) = 3: 12|3, 13|2, 1|23.
%e A361804 a(4) = 0.
%e A361804 a(5) = 15: 1234|5, 1235|4, 123|45, 1245|3, 124|35, 125|34, 12|345, 1345|2, 134|25, 135|24, 13|245, 145|23, 14|235, 15|234, 1|2345.
%e A361804 a(6) = 45: 12|34|5|6, 12|35|4|6, 12|3|45|6, 12|36|4|5, 12|3|46|5, 12|3|4|56, 13|24|5|6, 13|25|4|6, 13|2|45|6, 13|26|4|5, 13|2|46|5, 13|2|4|56, 14|23|5|6, 15|23|4|6, 1|23|45|6, 16|23|4|5, 1|23|46|5, 1|23|4|56, 14|25|3|6, 14|2|35|6, 14|26|3|5, 14|2|36|5, 14|2|3|56, 15|24|3|6, 1|24|35|6, 16|24|3|5, 1|24|36|5, 1|24|3|56, 15|2|34|6, 1|25|34|6, 16|2|34|5, 1|26|34|5, 1|2|34|56, 15|26|3|4, 15|2|36|4, 15|2|3|46, 16|25|3|4, 1|25|36|4, 1|25|3|46, 16|2|35|4, 1|26|35|4, 1|2|35|46, 16|2|3|45, 1|26|3|45, 1|2|36|45.
%p A361804 b:= proc(n, x, y) option remember; `if`(abs(x-y)>2*n, 0,
%p A361804      `if`(n=0, 1, b(n-1, x+1, y)+`if`(x>0, b(n-1, x-1, y+1)*x, 0)+
%p A361804      `if`(y>0, b(n-1, x+1, y-1)*y, 0)))
%p A361804     end:
%p A361804 a:= n-> b(n, 0$2):
%p A361804 seq(a(n), n=0..33);
%Y A361804 Cf. A000110, A003724, A004767, A004773, A005046, A275679.
%K A361804 nonn
%O A361804 0,4
%A A361804 _Alois P. Heinz_, Jun 12 2023