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A287672 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-9 is member of a block >= b-1.

%I A287672 #9 May 27 2018 10:33:45
%S A287672 1,1,2,5,15,52,203,877,4140,21147,115975,678570,4213596,27643917,
%T A287672 190873448,1382311874,10468191626,82676728870,679376431959,
%U A287672 5796059673820,51241781683988,468639956185787,4426928995172738,43131913020157751,432877526792148982
%N A287672 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-9 is member of a block >= b-1.
%H A287672 Alois P. Heinz, <a href="/A287672/b287672.txt">Table of n, a(n) for n = 0..30</a>
%H A287672 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%F A287672 a(n) = A287641(n,9).
%F A287672 a(n) = A000110(n) for n <= 11.
%e A287672 a(12) = 4213596 = 4213597 - 1 = A000110(12) - 1 counts all set partitions of [12] except: 13456789(10)(11)|2|(12).
%p A287672 b:= proc(n, l) option remember; `if`(n=0, 1, add(b(n-1,
%p A287672       [seq(max(l[i], j), i=2..nops(l)), j]), j=1..l[1]+1))
%p A287672     end:
%p A287672 a:= n-> b(n, [0$9]):
%p A287672 seq(a(n), n=0..20);
%t A287672 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 A287672 a[n_] := b[n, Table[0, 9]];
%t A287672 Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, May 27 2018, from Maple *)
%Y A287672 Column k=9 of A287641.
%Y A287672 Cf. A000110.
%K A287672 nonn
%O A287672 0,3
%A A287672 _Alois P. Heinz_, May 29 2017