A332942 Number of entries in the second blocks of all set partitions of [n] when blocks are ordered by increasing lengths.

1, 7, 25, 101, 366, 1555, 7099, 34627, 184033, 1059972, 6425992, 41266681, 280938451, 2009636335, 15025372685, 117386912433, 956458929950, 8104399834719, 71244441818927, 648761935841876, 6110827367541999, 59454153443971106, 596654820386392152

Offset: 2

Author

Alois P. Heinz, Mar 03 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 A319298.

Programs

Maple

b:= proc(n, i, t) option remember; `if`(n=0, [1, 0], `if`(i>n, 0,
      add((p-> p+`if`(t>0 and t-j<1, [0, p[1]*i], 0))(b(n-i*j, i+1,
      max(0, t-j))/j!*combinat[multinomial](n, i$j, n-i*j)), j=0..n/i)))
    end:
a:= n-> b(n, 1, 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 > n, 0 {0, 0}, Sum[ Function[p, p + If[t > 0 && t - j < 1, {0, p[[1]]*i}, {0, 0}]][b[n - i*j, i + 1, Max[0, t - j]]/j!*multinomial[n, Append[Table[i, {j}], n - i*j]]], {j, 0, n/i}]]];
a[n_] := b[n, 1, 2][[2]];
a /@ Range[2, 24] (* Jean-François Alcover, Jan 06 2021, after Alois P. Heinz *)