A339011 Sum over all partitions of n of the product of the number of parts and the number of distinct parts.

0, 1, 3, 8, 17, 34, 61, 107, 176, 284, 442, 676, 1007, 1483, 2140, 3055, 4299, 5993, 8255, 11284, 15272, 20529, 27373, 36274, 47735, 62484, 81293, 105251, 135555, 173818, 221836, 282003, 356980, 450256, 565765, 708537, 884296, 1100287, 1364736, 1687952, 2081724

Offset: 0

Author

Alois P. Heinz, Nov 18 2020

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..5000

Crossrefs

Cf. A096541, A339006, A339312.

Essentially partial sums of A093694.

Programs

Maple

b:= proc(n, i, p, d) option remember; `if`(n=0, d*p, `if`(i<1, 0,
      add(b(n-i*j, i-1, p+j, `if`(j=0, d, d+1)), j=0..n/i)))
    end:
a:= n-> b(n$2, 0$2):
seq(a(n), n=0..50);
# second Maple program:
b:= proc(n, i) option remember; `if`(n<=0 or i=0, [0$2],
     `if`(i=1, [1, n], b(n, i-1)+ (p-> p+[0, p[1]])(b(n-i, i))))
    end:
a:= proc(n) option remember; b(n$2)[2]+`if`(n<0, 0, a(n-1)) end:
seq(a(n), n=0..100);  # Alois P. Heinz, Jul 25 2022

Mathematica

b[n_, i_, p_, d_] := b[n, i, p, d] = If[n == 0, d*p, If[i < 1, 0,
     Sum[b[n - i*j, i - 1, p + j, If[j == 0, d, d + 1]], {j, 0, n/i}]]];
a[n_] := b[n, n, 0, 0];
a /@ Range[0, 50] (* Jean-François Alcover, Mar 09 2021, after Alois P. Heinz *)