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A287667 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-4 is member of a block >= b-1.

%I A287667 #10 May 27 2018 10:33:04
%S A287667 1,1,2,5,15,52,203,876,4116,20827,112538,645045,3900512,24769152,
%T A287667 164546915,1139818861,8209631792,61331709492,474221335902,
%U A287667 3787741281763,31199052157724,264605708064825,2307562757319104,20666169125398768,189855243829576499
%N A287667 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-4 is member of a block >= b-1.
%H A287667 Alois P. Heinz, <a href="/A287667/b287667.txt">Table of n, a(n) for n = 0..90</a>
%H A287667 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%F A287667 a(n) = A287641(n,4).
%F A287667 a(n) = A000110(n) for n <= 6.
%e A287667 a(7) = 876 = 877 - 1 = A000110(7) - 1 counts all set partitions of [7] except: 13456|2|7.
%p A287667 b:= proc(n, l) option remember; `if`(n=0, 1, add(b(n-1,
%p A287667       [seq(max(l[i], j), i=2..nops(l)), j]), j=1..l[1]+1))
%p A287667     end:
%p A287667 a:= n-> b(n, [0$4]):
%p A287667 seq(a(n), n=0..26);
%t A287667 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 A287667 a[n_] := b[n, Table[0, 4]];
%t A287667 Table[a[n], {n, 0, 26}] (* _Jean-François Alcover_, May 27 2018, from Maple *)
%Y A287667 Column k=4 of A287641.
%Y A287667 Cf. A000110.
%K A287667 nonn
%O A287667 0,3
%A A287667 _Alois P. Heinz_, May 29 2017