A333059 Number of entries in the second blocks of all set partitions of [n] when blocks are ordered by decreasing lengths.

1, 4, 17, 76, 357, 1737, 8997, 49420, 289253, 1793221, 11727861, 80576965, 579781009, 4356513727, 34118896917, 277963716808, 2351740613433, 20630800971825, 187374815249205, 1759353644746663, 17055176943817785, 170477858336708555, 1754992340756441973

Offset: 2

Author

Alois P. Heinz, Mar 06 2020

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 2..576
• Wikipedia, Partition of a set

Crossrefs

Column k=2 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, 2)[2]:
seq(a(n), n=2..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, 2][[2]];
a /@ Range[2, 24] (* Jean-François Alcover, Apr 24 2021, after Alois P. Heinz *)