A303586 - OEIS

%I #32 Jun 17 2024 06:58:48

%S 1,0,1,1,2,3,6,11,23,47,103,226,518,1200,2867,6946,17234,43393,111419,

%T 290242,768901,2065172,5630083,15549403,43527487,123343911,353864422,

%U 1026935904,3014535166,8945274505,26829206798,81293234754,248805520401,768882019073,2398686176048,7552071250781

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

%C a(n) appears to be the same as A211694(n-2). - _Georg Fischer_, Oct 09 2018

%H Alois P. Heinz, <a href="/A303586/b303586.txt">Table of n, a(n) for n = 0..1132</a>

%H A. O. Munagi, <a href="https://doi.org/10.1080/00029890.2018.1430960">Set partitions with isolated singletons</a>, Am. Math. Monthly 125 (2018), 447-452.

%p f:=proc(n) local j;

%p add(combinat:-bell(j-1)*binomial(n-j-1,j-1), j=0..floor(n/2));

%p end;

%p [seq(f(n),n=0..100)];

%t a[n_] := If[n == 0, 1, Sum[BellB[j-1]*Binomial[n-j-1, j-1], {j, 1, Floor[n/2]}]];

%t Table[a[n], {n, 0, 100}] (* _Jean-François Alcover_, Jun 17 2024, after Maple code *)

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

%K nonn

%O 0,5

%A _N. J. A. Sloane_, May 19 2018

%E a(0)=1 prepended by _Alois P. Heinz_, May 21 2023