A333061 Number of entries in the fourth blocks of all set partitions of [n] when blocks are ordered by decreasing lengths.

1, 11, 81, 512, 3151, 20071, 133853, 924320, 6551293, 47529561, 354259153, 2725545695, 21741995463, 180198265559, 1551865576121, 13865702570254, 128238585735637, 1224733005946425, 12053244176971825, 122035994844818345, 1269623551116437475, 13561114665253219451

Offset: 4

Author

Alois P. Heinz, Mar 06 2020

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 4..576
• Wikipedia, Partition of a set

Crossrefs

Column k=4 of A319375.

Programs

Maple

b:= proc(n, i, t) option remember; `if`(n=0, [1, 0], `if`(i<1, 0,
      add((p-> p+`if`(t>0 and t-j<1, [0, p[1]*i], 0))(
       combinat[multinomial](n, i$j, n-i*j)/j!*
      b(n-i*j, min(n-i*j, i-1), max(0, t-j))), j=0..n/i)))
    end:
a:= n-> b(n$2, 4)[2]:
seq(a(n), n=4..25);

Mathematica

multinomial[n_, k_List] := n!/Times @@ (k!);
b[n_, i_, t_] := b[n, i, t] = If[n == 0, {1, 0}, If[i < 1, {0, 0},
     Sum[Function[p, p + If[t > 0 && t - j < 1, {0, p[[1]]*i}, {0, 0}]][
     multinomial[n, Append[Table[i, {j}], n - i*j]]/j!*
     b[n - i*j, Min[n - i*j, i - 1], Max[0, t - j]]], {j, 0, n/i}]]];
a[n_] := b[n, n, 4][[2]];
a /@ Range[4, 25] (* Jean-François Alcover, Apr 24 2021, after Alois P. Heinz *)