A350683 Total sum over all partitions of [n] of elements i contained in block i when blocks are ordered with decreasing largest elements.

0, 1, 1, 8, 17, 98, 362, 1916, 9512, 53858, 315872, 1984979, 13105685, 91128546, 663815424, 5055622309, 40148341135, 331753228115, 2846786927873, 25323311882074, 233137061978065, 2218141402504254, 21780561656373552, 220451321425101091, 2297330116404668422

Offset: 0

Author

Alois P. Heinz, Jan 11 2022

Keywords

• nonn

Examples

			a(4) = 17 = 3*1 + 4*2 + 2*3: 432(1), 42(1)|3, 4(1)|3|2, 43|(2)1, 43|(2)|1, 4|3(2)1, 4|3(2)|1, 43(1)|(2), 4(1)|3(2).
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..575
• Wikipedia, Partition of a set

Crossrefs

Cf. A008805, A350648, A350684.

Programs

Maple

b:= proc(n, m) option remember; `if`(n=0, [1, 0], add((p-> [0,
     `if`(n=j, p[1]*j, 0)]+p)(b(n-1, max(m, j))), j=1..m+1))
    end:
a:= n-> b(n, 0)[2]:
seq(a(n), n=0..25);

Mathematica

b[n_, m_] := b[n, m] = If[n == 0, {1, 0}, Sum[With[{p = b[n-1, Max[m, j]]},
     {0, If[n == j, p[[1]]*j, 0]} + p], {j, 1, m+1}]];
a[n_] := b[n, 0][[2]];
Table[a[n], {n, 0, 25}]; (* Jean-François Alcover, May 08 2022, after Alois P. Heinz *)

Formula

a(n) = Sum_{k=1..max(0,A008805(n-1))} k * A350684(n,k).