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

2,2

Links

Alois P. Heinz, Table of n, a(n) for n = 2..576

Wikipedia, Partition of a set

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 *)

Crossrefs

Column k=2 of A319375.

Sequence in context: A239204 A005572 A202879 * A081922 A124325 A151248

Adjacent sequences: A333056 A333057 A333058 * A333060 A333061 A333062

Keyword

nonn

Author

Alois P. Heinz, Mar 06 2020

Status

approved