A350175 Sum of the distinct block sizes over all partitions of [n].

0, 1, 3, 13, 45, 196, 888, 4383, 22879, 129163, 768913, 4849912, 32202712, 224672241, 1640679589, 12517008985, 99484656169, 822410210044, 7055883373604, 62730142658947, 576984726864147, 5482889832932123, 53757450049841167, 543169144098559606, 5649499728403949184

Offset: 0

Author

Alois P. Heinz, Jan 06 2022

Keywords

• nonn

Examples

			a(3) = 13 = 1*3 + 3*(1+2) + 1: 123, 1|23, 13|2, 12|3, 1|2|3.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..400
• Wikipedia, Partition of a set

Crossrefs

Cf. A000110, A070071, A131719, A132963.

Programs

Maple

b:= proc(n, i, c) option remember; `if`(n=0, c,
      `if`(i<1, 0, add(b(n-j*i, i-1, c+i*signum(j))*
      combinat[multinomial](n, n-i*j, i$j)/j!, j=0..n/i)))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..30);

Mathematica

multinomial[n_, k_List] := n!/Times @@ (k!);
b[n_, i_, c_] := b[n, i, c] = If[n == 0, c,
     If[i < 1, 0, Sum[b[n - j*i, i - 1, c + i*Sign[j]]*
     multinomial[n, Join[{n - i*j}, Table[i, {j}]]]/j!, {j, 0, n/i}]]];
a[n_] := b[n, n, 0];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Jan 11 2022, after Alois P. Heinz *)

Formula

a(n) mod 2 = A131719(n).