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A363472 Total number of blocks in all partitions of [n] where each block has at least one odd element and at least one even element.

%I A363472 #29 Jun 05 2023 14:53:07
%S A363472 0,0,1,1,5,13,55,193,941,4081,22351,113761,694565,4030153,27107095,
%T A363472 175738753,1289775821,9209233921,73147903471,568928274961,
%U A363472 4857161139365,40796613003433,372190216061335,3352314486348433,32518958606637101,312271731474218881
%N A363472 Total number of blocks in all partitions of [n] where each block has at least one odd element and at least one even element.
%C A363472 All positive terms are odd.
%H A363472 Alois P. Heinz, <a href="/A363472/b363472.txt">Table of n, a(n) for n = 0..580</a>
%H A363472 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%F A363472 a(n) = Sum_{k=0..floor(n/2)} k * Stirling2(floor(n/2),k) * Stirling2(ceiling(n/2),k) * k!.
%e A363472 a(2) = 1: 12.
%e A363472 a(3) = 1: 123.
%e A363472 a(4) = 5 = 1 + 2 + 2: 1234, 12|34, 14|23.
%e A363472 a(5) = 13 = 1 + 2 + 2 + 2 + 2 + 2 + 2: 12345, 123|45, 125|34, 12|345, 134|25, 145|23, 14|235.
%p A363472 a:= n-> (h-> add(k*Stirling2(h, k)*Stirling2(n-h, k)*k!, k=0..h))(floor(n/2)):
%p A363472 seq(a(n), n=0..30);
%Y A363472 Cf. A124425, A362495, A363454.
%K A363472 nonn
%O A363472 0,5
%A A363472 _Alois P. Heinz_, Jun 05 2023