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A287671 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-8 is member of a block >= b-1.

%I A287671 #9 May 27 2018 10:33:38
%S A287671 1,1,2,5,15,52,203,877,4140,21147,115975,678569,4213333,27634757,
%T A287671 190697165,1379679500,10433619205,82253035850,674373619108,
%U A287671 5738060816421,50573749394877,460936356129618,4337525923676113,42084057817903853,420444371318055912
%N A287671 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-8 is member of a block >= b-1.
%H A287671 Alois P. Heinz, <a href="/A287671/b287671.txt">Table of n, a(n) for n = 0..33</a>
%H A287671 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%F A287671 a(n) = A287641(n,8).
%F A287671 a(n) = A000110(n) for n <= 10.
%e A287671 a(11) = 678569 = 678570 - 1 = A000110(11) - 1 counts all set partitions of [11] except: 13456789(10)|2|(11).
%p A287671 b:= proc(n, l) option remember; `if`(n=0, 1, add(b(n-1,
%p A287671       [seq(max(l[i], j), i=2..nops(l)), j]), j=1..l[1]+1))
%p A287671     end:
%p A287671 a:= n-> b(n, [0$8]):
%p A287671 seq(a(n), n=0..20);
%t A287671 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 A287671 a[n_] := b[n, Table[0, 8]];
%t A287671 Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, May 27 2018, from Maple *)
%Y A287671 Column k=8 of A287641.
%Y A287671 Cf. A000110.
%K A287671 nonn
%O A287671 0,3
%A A287671 _Alois P. Heinz_, May 29 2017