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A270954 Number of set partitions of [n] having no pairs (m,m+1) such that m+1 is in some block b and m is in block b+1.

%I A270954 #10 May 27 2018 07:00:20
%S A270954 1,1,2,4,9,25,84,323,1377,6412,32312,174941,1011357,6210298,40323011,
%T A270954 275763910,1979709852,14875239212,116679269248,953201694216,
%U A270954 8093501305721,71291395351760,650357174742217,6134966541625355,59759476024690454,600309156303711764
%N A270954 Number of set partitions of [n] having no pairs (m,m+1) such that m+1 is in some block b and m is in block b+1.
%H A270954 Alois P. Heinz, <a href="/A270954/b270954.txt">Table of n, a(n) for n = 0..300</a>
%H A270954 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%e A270954 a(3) = 4: 123, 12|3, 1|23, 1|2|3.
%e A270954 a(4) = 9: 1234, 123|4, 12|34, 12|3|4, 1|234, 1|23|4, 14|2|3, 1|2|34, 1|2|3|4.
%p A270954 b:= proc(n, i, m) option remember; `if`(n=0, 1, add(
%p A270954      `if`(j=i-1, 0, b(n-1, j, max(m, j))), j=1..m+1))
%p A270954     end:
%p A270954 a:= n-> b(n, 1, 0):
%p A270954 seq(a(n), n=0..25);
%t A270954 b[n_, i_, m_] := b[n, i, m] = If[n == 0, 1, Sum[If[j == i - 1, 0, b[n - 1, j, Max[m, j]]], {j, 1, m + 1}]];
%t A270954 a[n_] := b[n, 1, 0];
%t A270954 Table[a[n], {n, 0, 25}] (* _Jean-François Alcover_, May 27 2018, translated from Maple *)
%Y A270954 Column k=0 of A270953.
%K A270954 nonn
%O A270954 0,3
%A A270954 _Alois P. Heinz_, Mar 26 2016