login
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, 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
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
KEYWORD
nonn
AUTHOR
Alois P. Heinz, May 17 2023
STATUS
approved