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A274837 Number of set partitions of [n] such that the difference between each element and its block index is a multiple of four.

%I A274837 #9 May 15 2018 08:34:29
%S A274837 1,1,1,1,1,2,3,4,5,11,22,41,72,191,459,1033,2209,6696,18777,49526,
%T A274837 124011,419203,1329966,4009931,11544970,43203329,152247581,511143253,
%U A274837 1644388769,6707557342,25952578959,95992345048,340793163873,1501194339387,6305017609678
%N A274837 Number of set partitions of [n] such that the difference between each element and its block index is a multiple of four.
%H A274837 Alois P. Heinz, <a href="/A274837/b274837.txt">Table of n, a(n) for n = 0..768</a>
%H A274837 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%e A274837 a(7) = 4: 15|26|37|4, 1|26|37|4|5, 1|2|37|4|5|6, 1|2|3|4|5|6|7.
%e A274837 a(8) = 5: 15|26|37|48, 1|26|37|48|5, 1|2|37|48|5|6, 1|2|3|48|5|6|7, 1|2|3|4|5|6|7|8.
%e A274837 a(9) = 11: 159|26|37|48, 15|26|37|48|9, 19|26|37|48|5, 1|26|37|48|59, 19|2|37|48|5|6, 1|2|37|48|59|6, 19|2|3|48|5|6|7, 1|2|3|48|59|6|7, 19|2|3|4|5|6|7|8, 1|2|3|4|59|6|7|8, 1|2|3|4|5|6|7|8|9.
%p A274837 b:= proc(n, m, t) option remember; `if`(n=0, 1,
%p A274837      add(`if`(irem(j-t, 4)=0, b(n-1, max(m, j),
%p A274837               irem(t+1, 4)), 0), j=1..m+1))
%p A274837     end:
%p A274837 a:= n-> b(n, 0, 1):
%p A274837 seq(a(n), n=0..35);
%t A274837 b[n_, m_, t_] := b[n, m, t] = If[n == 0, 1, Sum[If[Mod[j - t, 4] == 0, b[n - 1, Max[m, j], Mod[t + 1, 4]], 0], {j, 1, m + 1}]];
%t A274837 a[n_] := b[n, 0, 1];
%t A274837 Table[a[n], {n, 0, 35}] (* _Jean-François Alcover_, May 15 2018, after _Alois P. Heinz_ *)
%Y A274837 Column k=4 of A274835.
%K A274837 nonn
%O A274837 0,6
%A A274837 _Alois P. Heinz_, Jul 08 2016