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

3,2

Links

Alois P. Heinz, Table of n, a(n) for n = 3..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, 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 *)

Crossrefs

Column k=3 of A319375.

Sequence in context: A037538 A037482 A147546 * A331719 A020085 A356076

Adjacent sequences: A333057 A333058 A333059 * A333061 A333062 A333063

Keyword

nonn

Author

Alois P. Heinz, Mar 06 2020

Status

approved