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A362495 Total number of blocks containing at least one odd element and at least one even element in all partitions of [n].

%I A362495 #66 Jun 05 2023 10:43:38
%S A362495 0,0,1,3,13,54,262,1294,7109,40367,248651,1587414,10827740,76494630,
%T A362495 571499993,4414720825,35798107309,299547765240,2616358573834,
%U A362495 23536296521084,220030456297349,2114721297588097,21046291460160803,214984439282684504,2267305399918683232
%N A362495 Total number of blocks containing at least one odd element and at least one even element in all partitions of [n].
%H A362495 Alois P. Heinz, <a href="/A362495/b362495.txt">Table of n, a(n) for n = 0..575</a>
%H A362495 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%F A362495 a(n) = Sum_{k=0..floor(n/2)} k * A124418(n,k).
%F A362495 a(n) = A138378(n) - A363434(n) = A005493(n-1) - A363434(n) for n>=1.
%e A362495 a(3) = 3 = 1 + 1 + 0 + 1 + 0 : 123, 12|3, 13|2, 1|23, 1|2|3.
%p A362495 b:= proc(n, x, y, m) option remember; `if`(n=0, m,
%p A362495       `if`(x+m>0, b(n-1, y, x, m)*(x+m), 0)+b(n-1, y, x+1, m)+
%p A362495       `if`(y>0, b(n-1, y-1, x, m+1)*y, 0))
%p A362495     end:
%p A362495 a:= n-> b(n, 0$3):
%p A362495 seq(a(n), n=0..25);
%Y A362495 Cf. A005493, A124418, A124425, A138378, A363434.
%K A362495 nonn
%O A362495 0,4
%A A362495 _Alois P. Heinz_, Jun 05 2023