A209536 Number of partitions of 0 having positive part-sum <= n.

1, 5, 14, 39, 88, 209, 434, 918, 1818, 3582, 6718, 12647, 22848, 41073, 72049, 125410, 213619, 361844, 601944, 995073, 1622337, 2626341, 4201366, 6681991, 10515755, 16449851, 25509951, 39333475, 60172700, 91577516, 138390480, 208096281, 310976730, 462512830

Offset: 1

Author

Clark Kimberling, Mar 10 2012

Keywords

• nonn

Comments

A partition of 0 is a set {i(1), i(2),..., i(n)} of nonzero integers with sum 0. Such a set uniquely partitions into two multisets {x(1),..., x(j)} and {y(1),..., y(k)} where x(1)+x(2)+...+x(j) =-[y(1)+y(2)+...+y(k)] and x(i) > 0 and y(i) < 0 for every i. The number x(1)+x(2)+...+x(j) is the positive part-sum.

Let p(h) be the number of partitions of h>=1, as in A000041. There are p(h)^2 ways to choose each of the sets {x(1),...,x(j)} and {y(1),...,y(k)} having sum h. Consequently, there are p(1)^2+p(2)^2+...+p(n)^2 partitions of 0 having positive part-sum <= n.

Examples

			0 = 1-1 = 2-2 = 2-(1+1) = (1+1)-2 = (1+1)-(1+1),
so that a(2) = 5.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..5000

Crossrefs

Cf. A209535, A000041, A259399.

Programs

Maple

a:= proc(n) option remember; `if`(n=0, 0,
      combinat[numbpart](n)^2+a(n-1))
    end:
seq(a(n), n=1..40);  # Alois P. Heinz, Oct 21 2018

Mathematica

p[n_] := IntegerPartitions[n]; l[n_] := Length[p[n]];
s[n_] := Sum[l[k]^2, {k, 1, n}];
Table[s[n], {n, 1, 40}] (* A209536 *)
(* Second program: *)
a[n_] := a[n] = If[n == 0, 0, PartitionsP[n]^2 + a[n-1]];
Array[a, 40] (* Jean-François Alcover, Jun 09 2021, after Alois P. Heinz *)

Formula

From Alois P. Heinz, Oct 21 2018: (Start)

a(n) = Sum_{j=1..n} A000041(j)^2.

a(n) = -1 + A259399(n). (End)