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A287669 Number of set partitions of [n] such that j is member of block b only if b = 1 or at least one of j-1, ..., j-6 is member of a block >= b-1.

%I A287669 #12 May 27 2018 10:33:14
%S A287669 1,1,2,5,15,52,203,877,4140,21146,115903,677026,4190648,27356008,
%T A287669 187573260,1346289439,10084570537,78630320221,636692795555,
%U A287669 5342949225111,46381106554291,415803352327861,3843867571153341,36592205230965683,358266592635074429
%N A287669 Number of set partitions of [n] such that j is member of block b only if b = 1 or at least one of j-1, ..., j-6 is member of a block >= b-1.
%H A287669 Alois P. Heinz, <a href="/A287669/b287669.txt">Table of n, a(n) for n = 0..45</a>
%H A287669 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%F A287669 a(n) = A287641(n,6).
%F A287669 a(n) = A000110(n) for n <= 8.
%e A287669 a(9) = 21146 = 21147 - 1 = A000110(9) - 1 counts all set partitions of [9] except: 1345678|2|9.
%p A287669 b:= proc(n, l) option remember; `if`(n=0, 1, add(b(n-1,
%p A287669       [seq(max(l[i], j), i=2..nops(l)), j]), j=1..l[1]+1))
%p A287669     end:
%p A287669 a:= n-> b(n, [0$6]):
%p A287669 seq(a(n), n=0..20);
%t A287669 b[n_, l_] := b[n, l] = If[n == 0, 1, Sum[b[n - 1, Append[Table[Max[l[[i]], j], {i, 2, Length[l]}], j]], {j, 1, l[[1]] + 1}]];
%t A287669 a[n_] := b[n, Table[0, 6]];
%t A287669 Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, May 27 2018, from Maple *)
%Y A287669 Column k=6 of A287641.
%Y A287669 Cf. A000110.
%K A287669 nonn
%O A287669 0,3
%A A287669 _Alois P. Heinz_, May 29 2017