A272065 - OEIS

%I #8 Feb 03 2017 14:51:17

%S 0,0,0,0,2,17,101,545,2935,16351,95335,583373,3745903,25208633,

%T 177505205,1305468285,10009943248,79880835800,662319435622,

%U 5696570446421,50749156111271,467630493212126,4451067568592918,43709810099960739,442331477265626019

%N 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 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>

%F a(n) = A000110(n) - A272064(n).

%e a(4) = 2: 13|24, 13|2|4.

%e 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 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-> combinat[bell](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_] := BellB[n]-b[n, 0, 0, {}]; Table[a[n], {n, 0, 18}] (* _Jean-François Alcover_, Feb 03 2017, translated from Maple *)

%Y Cf. A000110, A185982, A271271, A272064.

%K nonn

%O 0,5

%A _Alois P. Heinz_, Apr 19 2016