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A272065 Number of set partitions of [n] such that at least one pair of consecutive blocks (b,b+1) exists having not exactly one pair of consecutive numbers (i,i+1) with i member of b and i+1 member of b+1.

%I A272065 #8 Feb 03 2017 14:51:17
%S A272065 0,0,0,0,2,17,101,545,2935,16351,95335,583373,3745903,25208633,
%T A272065 177505205,1305468285,10009943248,79880835800,662319435622,
%U A272065 5696570446421,50749156111271,467630493212126,4451067568592918,43709810099960739,442331477265626019
%N A272065 Number of set partitions of [n] such that at least one pair of consecutive blocks (b,b+1) exists having not exactly one pair of consecutive numbers (i,i+1) with i member of b and i+1 member of b+1.
%H A272065 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%F A272065 a(n) = A000110(n) - A272064(n).
%e A272065 a(4) = 2: 13|24, 13|2|4.
%e A272065 a(5) = 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 A272065 b:= proc(n, i, m, l) option remember; `if`(n=0,
%p A272065       `if`({l[], 1}={1}, 1, 0), add(`if`(j<m+1 and
%p A272065            j=i+1 and l[j]=1, 0, b(n-1, j, max(m, j),
%p A272065       `if`(j=m+1, [l[], `if`(j=i+1, 1, 0)],
%p A272065       `if`(j=i+1, subsop(j=1, l), l)))), j=1..m+1))
%p A272065     end:
%p A272065 a:= n-> combinat[bell](n)-b(n, 0$2, []):
%p A272065 seq(a(n), n=0..18);
%t A272065 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_] := BellB[n]-b[n, 0, 0, {}]; Table[a[n], {n, 0, 18}] (* _Jean-François Alcover_, Feb 03 2017, translated from Maple *)
%Y A272065 Cf. A000110, A185982, A271271, A272064.
%K A272065 nonn
%O A272065 0,5
%A A272065 _Alois P. Heinz_, Apr 19 2016