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.

1, 1, 0, 0, 1, 2, 0, 0, 20, 48, 0, 0, 1147, 3968, 0, 0, 173203, 709488, 0, 0, 53555964, 246505600, 0, 0, 28368601065, 148963383616, 0, 0, 24044155851601, 141410718244864, 0, 0, 30934515698084780, 198914201874983936, 0, 0, 57215369885233295955, 398742900995358584320

Offset

0,6

Comments

All odd elements are in blocks with an odd block size and all even elements are in blocks with an even block size.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..672

Wikipedia, Partition of a set

Formula

a(n) = A003724(ceiling(n/2)) * A005046(floor(n/4)) if (n mod 4) in {0,1}.

a(n) = 0 if (n mod 4) in {2,3}.

Example

a(0) = 1: (), the empty partition.
a(1) = 1: 1.
a(4) = 1: 1|24|3.
a(5) = 2: 135|24, 1|24|3|5.
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.

Maple

b:= proc(n, t) option remember; `if`(n=0, 1, add(
     `if`((j+t)::even, b(n-j, t)*binomial(n-1, j-1), 0), j=1..n))
    end:
a:= n-> (h-> b(n-h, 1)*b(h, 0))(iquo(n, 2)):
seq(a(n), n=0..40);

Mathematica

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}]];
a[n_] := b[n - #, 1]*b[#, 0]&[Quotient[n, 2]];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Nov 18 2023, after Alois P. Heinz *)

Crossrefs

Cf. A000110, A003724, A005046, A124419, A274538, A275679.

Sequence in context: A368849 A244143 A066294 * A230840 A209454 A287713

Adjacent sequences: A363070 A363071 A363072 * A363074 A363075 A363076

Keyword

nonn

Author

Alois P. Heinz, May 17 2023

Status

approved