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A211694 Number of partitions of [n] that contain no isolated singletons.

%I A211694 #19 Feb 07 2025 10:36:51
%S A211694 1,0,1,1,2,3,6,11,23,47,103,226,518,1200,2867,6946,17234,43393,111419,
%T A211694 290242,768901,2065172,5630083,15549403,43527487,123343911,353864422,
%U A211694 1026935904,3014535166,8945274505,26829206798,81293234754,248805520401,768882019073,2398686176048,7552071250781
%N A211694 Number of partitions of [n] that contain no isolated singletons.
%C A211694 Number of nonnegative integer arrays of length n with new values introduced in order 0 upwards and every value appearing only in runs of at least 2.
%C A211694 Column 2 of A211700.
%H A211694 Alois P. Heinz, <a href="/A211694/b211694.txt">Table of n, a(n) for n = 0..1132</a>
%H A211694 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.
%F A211694 G.f.: 1+x^2/W(0), where W(k) = 1 - x - x^2/(1 - x^2*(k+1)/W(k+1) ); (continued fraction). - _Sergei N. Gladkovskii_, Nov 10 2014
%e A211694 All solutions for n = 7:
%e A211694   0    0    0    0    0    0    0    0    0    0    0
%e A211694   0    0    0    0    0    0    0    0    0    0    0
%e A211694   0    0    1    0    1    0    0    0    1    1    1
%e A211694   1    1    1    1    1    0    0    0    1    1    1
%e A211694   1    1    2    1    1    0    1    0    0    1    1
%e A211694   2    1    2    0    2    0    1    1    0    1    0
%e A211694   2    1    2    0    2    0    1    1    0    1    0
%p A211694 f:=proc(n) local j;
%p A211694 add(combinat:-bell(j-1)*binomial(n-j-1, j-1), j=0..floor(n/2));
%p A211694 end;
%p A211694 [seq(f(n), n=0..100)]; # _N. J. A. Sloane_, May 19 2018
%t A211694 a[n_] := If[n == 0, 1, Sum[BellB[j-1]*Binomial[n-j-1, j-1], {j, 1, Floor[n/2]}]];
%t A211694 Table[a[n], {n, 0, 100}] (* _Jean-François Alcover_, Jun 17 2024, after Maple code *)
%Y A211694 Cf. A160823, A211700.
%Y A211694 Cf. A198648, A198655, A303587, A303588, A363181.
%K A211694 nonn
%O A211694 0,5
%A A211694 _R. H. Hardin_, Apr 19 2012
%E A211694 Edited by _Andrey Zabolotskiy_, Feb 07 2025