A271764 Number of set partitions of [n] with minimal block length multiplicity equal to four.

1, 0, 0, 0, 105, 0, 0, 0, 67375, 135135, 1261260, 675675, 50925875, 97847750, 703993290, 6215737710, 228687298476, 58017429575, 11262925616250, 72813288304295, 2841531210935725, 11311740884766630, 252469888906590355, 2207276997956560530, 28579415631325499655

Offset: 4

Author

Alois P. Heinz, Apr 13 2016

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 4..577
• Wikipedia, Partition of a set

Crossrefs

Column k=4 of A271424.

Programs

Maple

with(combinat):
b:= proc(n, i, k) option remember; `if`(n=0, 1,
      `if`(i<1, 0, add(multinomial(n, n-i*j, i$j)
        *b(n-i*j, i-1, k)/j!, j={0, $k..n/i})))
    end:
a:= n-> b(n$2, 4)-b(n$2, 5):
seq(a(n), n=4..30);

Mathematica

multinomial[n_, k_List] := n!/Times @@ (k!);
b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[multinomial[n, Join[{n - i*j}, Table[i, j]]]*b[n - i*j, i - 1, k]/j!, {j, Join[{0}, Range[k, n/i]]}]]];
a[n_] := b[n, n, 4] - b[n, n, 5];
Table[a[n], {n, 4, 30}] (* Jean-François Alcover, May 15 2018, after Alois P. Heinz *)

Formula

a(n) = A271424(n,4).