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A274839 Number of set partitions of [n] such that the difference between each element and its block index is a multiple of six.

%I A274839 #11 May 15 2018 08:34:45
%S A274839 1,1,1,1,1,1,1,2,3,4,5,6,7,15,30,57,104,183,310,811,1939,4377,9497,
%T A274839 19987,40883,121620,339817,902822,2301883,5665060,13489425,44503335,
%U A274839 140080438,425111779,1250942834,3575716011,9910354002,36376567529,127026151621
%N A274839 Number of set partitions of [n] such that the difference between each element and its block index is a multiple of six.
%H A274839 Alois P. Heinz, <a href="/A274839/b274839.txt">Table of n, a(n) for n = 0..842</a>
%H A274839 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%e A274839 a(7) = 2: 17|2|3|4|5|6, 1|2|3|4|5|6|7.
%e A274839 a(8) = 3: 17|28|3|4|5|6, 1|28|3|4|5|6|7, 1|2|3|4|5|6|7|8.
%e A274839 a(9) = 4: 17|28|39|4|5|6, 1|28|39|4|5|6|7, 1|2|39|4|5|6|7|8, 1|2|3|4|5|6|7|8|9.
%p A274839 b:= proc(n, m, t) option remember; `if`(n=0, 1,
%p A274839      add(`if`(irem(j-t, 6)=0, b(n-1, max(m, j),
%p A274839               irem(t+1, 6)), 0), j=1..m+1))
%p A274839     end:
%p A274839 a:= n-> b(n, 0, 1):
%p A274839 seq(a(n), n=0..40);
%t A274839 b[n_, m_, t_] := b[n, m, t] = If[n == 0, 1, Sum[If[Mod[j - t, 6] == 0, b[n - 1, Max[m, j], Mod[t + 1, 6]], 0], {j, 1, m + 1}]];
%t A274839 a[n_] := b[n, 0, 1];
%t A274839 Table[a[n], {n, 0, 40}] (* _Jean-François Alcover_, May 15 2018, after _Alois P. Heinz_ *)
%Y A274839 Column k=6 of A274835.
%K A274839 nonn
%O A274839 0,8
%A A274839 _Alois P. Heinz_, Jul 08 2016