A322876 Number of set partitions of [n] such that the maximal absolute difference between the least elements of consecutive blocks equals three.

0, 1, 7, 39, 209, 1123, 6153, 34723, 202852, 1229672, 7742792, 50653678, 344195782, 2427812876, 17761759538, 134650690097, 1056676856777, 8574943334545, 71881479393513, 621792661601615, 5544644720281979, 50918125911279963, 481093310682127190

Offset: 3

Author

Alois P. Heinz, Dec 29 2018

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 3..586
• Wikipedia, Partition of a set

Crossrefs

Column k=3 of A287215.

Cf. A287252, A287253.

Programs

Maple

b:= proc(n, k, m, l) option remember; `if`(n<1, 1,
     `if`(l-n>k, 0, b(n-1, k, m+1, n))+m*b(n-1, k, m, l))
    end:
A:= (n, k)-> b(n-1, min(k, n-1), 1, n):
a:= n-> (k-> A(n, k)-A(n, k-1))(3):
seq(a(n), n=3..30);

Mathematica

b[n_, k_, m_, l_] := b[n, k, m, l] = If[n < 1, 1, If[l - n > k, 0, b[n - 1, k, m + 1, n]] + m b[n - 1, k, m, l]];
A[n_, k_] := b[n - 1, Min[k, n - 1], 1, n];
a[n_] := With[{k = 3}, A[n, k] - A[n, k - 1]];
a /@ Range[3, 30] (* Jean-François Alcover, May 05 2020, after Maple *)

Formula

a(n) = A287253(n) - A287252(n).