A274840 Number of set partitions of [n] such that the difference between each element and its block index is a multiple of seven.

1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 8, 17, 34, 65, 120, 215, 374, 629, 1641, 3919, 8849, 19241, 40707, 84203, 170229, 503902, 1407019, 3746718, 9600121, 23815820, 57408783, 134592586, 440661179, 1383544922, 4206645985, 12456581554, 36012285385

Offset: 0

Author

Alois P. Heinz, Jul 08 2016

Keywords

• nonn

Examples

			a(8) = 2: 18|2|3|4|5|6|7, 1|2|3|4|5|6|7|8.
a(9) = 3: 18|29|3|4|5|6|7, 1|29|3|4|5|6|7|8, 1|2|3|4|5|6|7|8|9.
a(10) = 4: 18|29|3(10)|4|5|6|7, 1|29|3(10)|4|5|6|7|8, 1|2|3(10)|4|5|6|7|8|9, 1|2|3|4|5|6|7|8|9|(10).
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..874
• Wikipedia, Partition of a set

Crossrefs

Column k=7 of A274835.

Programs

Maple

b:= proc(n, m, t) option remember; `if`(n=0, 1,
     add(`if`(irem(j-t, 7)=0, b(n-1, max(m, j),
              irem(t+1, 7)), 0), j=1..m+1))
    end:
a:= n-> b(n, 0, 1):
seq(a(n), n=0..42);

Mathematica

b[n_, m_, t_] := b[n, m, t] = If[n == 0, 1, Sum[If[Mod[j - t, 7] == 0, b[n - 1, Max[m, j], Mod[t + 1, 7]], 0], {j, 1, m + 1}]];
a[n_] := b[n, 0, 1];
Table[a[n], {n, 0, 42}] (* Jean-François Alcover, May 15 2018, after Alois P. Heinz *)