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A287670 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-7 is member of a block >= b-1.

%I A287670 #9 May 27 2018 10:33:25
%S A287670 1,1,2,5,15,52,203,877,4140,21147,115974,678434,4209827,27578206,
%T A287670 189954361,1370870811,10334533723,81166980407,662588540048,
%U A287670 5610196619724,49177794178940,445536788068643,4165402700226511,40131393651398259,397935154986242021
%N A287670 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-7 is member of a block >= b-1.
%H A287670 Alois P. Heinz, <a href="/A287670/b287670.txt">Table of n, a(n) for n = 0..37</a>
%H A287670 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%F A287670 a(n) = A287641(n,7).
%F A287670 a(n) = A000110(n) for n <= 9.
%e A287670 a(10) = 115974 = 115975 - 1 = A000110(10) - 1 counts all set partitions of [10] except: 13456789|2|(10).
%p A287670 b:= proc(n, l) option remember; `if`(n=0, 1, add(b(n-1,
%p A287670       [seq(max(l[i], j), i=2..nops(l)), j]), j=1..l[1]+1))
%p A287670     end:
%p A287670 a:= n-> b(n, [0$7]):
%p A287670 seq(a(n), n=0..20);
%t A287670 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 A287670 a[n_] := b[n, Table[0, 7]];
%t A287670 Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, May 27 2018, from Maple *)
%Y A287670 Column k=7 of A287641.
%Y A287670 Cf. A000110.
%K A287670 nonn
%O A287670 0,3
%A A287670 _Alois P. Heinz_, May 29 2017