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

0,3

Links

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

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 *)

Crossrefs

Cf. A096541, A339006, A339312.

Essentially partial sums of A093694.

Sequence in context: A281166 A165273 A002626 * A029859 A344004 A305105

Adjacent sequences: A339008 A339009 A339010 * A339012 A339013 A339014

Keyword

nonn

Author

Alois P. Heinz, Nov 18 2020

Status

approved