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%I #11 Feb 03 2017 14:50:48
%S 1,1,2,5,13,35,102,332,1205,4796,20640,95197,467694,2435804,13394117,
%T 77490260,470198899,2984034004,19757370537,136171758636,975002124101,
%U 7239322944625,55648169854405,442195755123607,3627392029179270,30679238282421267,267215329668444337
%N Number of set partitions of [n] such that for each pair of consecutive blocks (b,b+1) exactly one pair of consecutive numbers (i,i+1) exists with i member of b and i+1 member of b+1.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%F a(n) = A000110(n) - A272065(n).
%e A000110(4) - a(4) = 15 - 13 = 2: 13|24, 13|2|4.
%e A000110(5) - a(5) = 52 - 35 = 17: 124|35, 124|3|5, 134|25, 134|2|5, 135|24, 13|245, 13|24|5, 135|2|4, 13|25|4, 13|2|45, 13|2|4|5, 14|235, 14|23|5, 14|25|3, 14|2|3|5, 1|24|35, 1|24|3|5.
%p b:= proc(n, i, m, l) option remember; `if`(n=0,
%p `if`({l[], 1}={1}, 1, 0), add(`if`(j<m+1 and
%p j=i+1 and l[j]=1, 0, b(n-1, j, max(m, j),
%p `if`(j=m+1, [l[], `if`(j=i+1, 1, 0)],
%p `if`(j=i+1, subsop(j=1, l), l)))), j=1..m+1))
%p end:
%p a:= n-> b(n, 0$2, []):
%p seq(a(n), n=0..18);
%t b[n_, i_, m_, l_] := b[n, i, m, l] = If[n==0, If[Union[Append[l, 1]] == {1}, 1, 0], Sum[If[j<m+1 && j==i+1 && l[[j]]==1, 0, b[n-1, j, Max[m, j], If[j==m+1, Append[l, If[j==i+1, 1, 0]], If[j==i+1, ReplacePart[l, j -> 1], l]]]], {j, 1, m+1}]]; a[n_] := b[n, 0, 0, {}]; Table[a[n], {n, 0, 18}] (* _Jean-François Alcover_, Feb 03 2017, translated from Maple *)
%Y Cf. A000110, A185982, A271270, A272065.
%K nonn
%O 0,3
%A _Alois P. Heinz_, Apr 19 2016
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