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A363073 Number of set partitions of [n] such that each element is contained in a block whose block size parity coincides with the parity of the element.

%I A363073 #37 Nov 18 2023 04:02:20
%S A363073 1,1,0,0,1,2,0,0,20,48,0,0,1147,3968,0,0,173203,709488,0,0,53555964,
%T A363073 246505600,0,0,28368601065,148963383616,0,0,24044155851601,
%U A363073 141410718244864,0,0,30934515698084780,198914201874983936,0,0,57215369885233295955,398742900995358584320
%N A363073 Number of set partitions of [n] such that each element is contained in a block whose block size parity coincides with the parity of the element.
%C A363073 All odd elements are in blocks with an odd block size and all even elements are in blocks with an even block size.
%H A363073 Alois P. Heinz, <a href="/A363073/b363073.txt">Table of n, a(n) for n = 0..672</a>
%H A363073 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%F A363073 a(n) = A003724(ceiling(n/2)) * A005046(floor(n/4)) if (n mod 4) in {0,1}.
%F A363073 a(n) = 0 if (n mod 4) in {2,3}.
%e A363073 a(0) = 1: (), the empty partition.
%e A363073 a(1) = 1: 1.
%e A363073 a(4) = 1: 1|24|3.
%e A363073 a(5) = 2: 135|24, 1|24|3|5.
%e A363073 a(8) = 20: 135|2468|7, 135|24|68|7, 137|2468|5, 137|24|5|68, 135|26|48|7, 135|28|46|7, 137|26|48|5, 137|28|46|5, 157|2468|3, 157|24|3|68, 1|2468|357, 1|24|357|68, 1|2468|3|5|7, 1|24|3|5|68|7, 157|26|3|48, 157|28|3|46, 1|26|357|48, 1|28|357|46, 1|26|3|48|5|7, 1|28|3|46|5|7.
%p A363073 b:= proc(n, t) option remember; `if`(n=0, 1, add(
%p A363073      `if`((j+t)::even, b(n-j, t)*binomial(n-1, j-1), 0), j=1..n))
%p A363073     end:
%p A363073 a:= n-> (h-> b(n-h, 1)*b(h, 0))(iquo(n, 2)):
%p A363073 seq(a(n), n=0..40);
%t A363073 b[n_, t_] := b[n, t] = If[n == 0, 1, Sum[If[EvenQ[j + t], b[n - j, t]* Binomial[n - 1, j - 1], 0], {j, 1, n}]];
%t A363073 a[n_] := b[n - #, 1]*b[#, 0]&[Quotient[n, 2]];
%t A363073 Table[a[n], {n, 0, 40}] (* _Jean-François Alcover_, Nov 18 2023, after _Alois P. Heinz_ *)
%Y A363073 Cf. A000110, A003724, A005046, A124419, A274538, A275679.
%K A363073 nonn
%O A363073 0,6
%A A363073 _Alois P. Heinz_, May 17 2023