A303586 Number of partitions of [n] that contain no isolated singletons.

1, 0, 1, 1, 2, 3, 6, 11, 23, 47, 103, 226, 518, 1200, 2867, 6946, 17234, 43393, 111419, 290242, 768901, 2065172, 5630083, 15549403, 43527487, 123343911, 353864422, 1026935904, 3014535166, 8945274505, 26829206798, 81293234754, 248805520401, 768882019073, 2398686176048, 7552071250781

Offset

0,5

Comments

a(n) appears to be the same as A211694(n-2). - Georg Fischer, Oct 09 2018

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1132

A. O. Munagi, Set partitions with isolated singletons, Am. Math. Monthly 125 (2018), 447-452.

Maple

f:=proc(n) local j;
add(combinat:-bell(j-1)*binomial(n-j-1, j-1), j=0..floor(n/2));
end;
[seq(f(n), n=0..100)];

Mathematica

a[n_] := If[n == 0, 1, Sum[BellB[j-1]*Binomial[n-j-1, j-1], {j, 1, Floor[n/2]}]];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Jun 17 2024, after Maple code *)

Crossrefs

Cf. A198648, A198655, A211694, A303587, A303588, A363181.

Sequence in context: A198648 A198655 A211694 * A130131 A338355 A277796

Adjacent sequences: A303583 A303584 A303585 * A303587 A303588 A303589

Keyword

nonn

Author

N. J. A. Sloane, May 19 2018

Extensions

a(0)=1 prepended by Alois P. Heinz, May 21 2023

Status

approved