A333060 Number of entries in the third blocks of all set partitions of [n] when blocks are ordered by decreasing lengths.

1, 7, 36, 186, 1023, 5867, 34744, 211888, 1343046, 8896185, 61801182, 449917898, 3425580850, 27183592435, 224196765392, 1917038645772, 16963064269986, 155112925687673, 1464150720422785, 14253033440621462, 142967758696293317, 1476398153663677539

Offset: 3

Author

Alois P. Heinz, Mar 06 2020

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 3..576
• Wikipedia, Partition of a set

Crossrefs

Column k=3 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, 3)[2]:
seq(a(n), n=3..24);

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, 3][[2]];
a /@ Range[3, 24] (* Jean-François Alcover, Apr 24 2021, after Alois P. Heinz *)