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A287673 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-10 is member of a block >= b-1.

%I A287673 #9 May 27 2018 10:32:36
%S A287673 1,1,2,5,15,52,203,877,4140,21147,115975,678570,4213597,27644436,
%T A287673 190898290,1382887161,10477990158,82819430415,681282289857,
%U A287673 5820296183791,51541816775857,472306124149579,4471549108520595,43676154621078016,439558508006341652
%N A287673 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-10 is member of a block >= b-1.
%H A287673 Alois P. Heinz, <a href="/A287673/b287673.txt">Table of n, a(n) for n = 0..29</a>
%H A287673 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%F A287673 a(n) = A287641(n,10).
%F A287673 a(n) = A000110(n) for n <= 12.
%e A287673 a(13) = 27644436 = 27644437 - 1 = A000110(13) - 1 counts all set partitions of [13] except: 13456789(10)(11)(12)|2|(13).
%p A287673 b:= proc(n, l) option remember; `if`(n=0, 1, add(b(n-1,
%p A287673       [seq(max(l[i], j), i=2..nops(l)), j]), j=1..l[1]+1))
%p A287673     end:
%p A287673 a:= n-> b(n, [0$10]):
%p A287673 seq(a(n), n=0..20);
%t A287673 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 A287673 a[n_] := b[n, Table[0, 10]];
%t A287673 Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, May 27 2018, from Maple *)
%Y A287673 Column k=10 of A287641.
%Y A287673 Cf. A000110.
%K A287673 nonn
%O A287673 0,3
%A A287673 _Alois P. Heinz_, May 29 2017