0,4

The first element of each partition is given weight 0.

Consider the partitions of n, e.g., n=5. For each partition sum T(e-1) and sum all these. E.g., 5 -> T(4)=10, 41 -> T(3)+T(0)=6, 32 -> T(2)+T(1)=4, 311 -> T(2)+T(0)+T(0)=3, 221 -> T(1)+T(1)+T(0)=2, 21111 ->1 and 11111 ->0. Summing, 10+6+4+3+2+1+0 = 26 as desired. - Jon Perry, Dec 12 2003

Also equals the sum of f(p) over the partitions p of n, where f(p) is obtained by replacing each part p_i of partition p by p_i*(p_i-1)/2. See I. G. Macdonald: Symmetric functions and Hall polynomials 2nd edition, p. 3, eqn (1.5) and (1.6). - Wouter Meeussen, Sep 25 2014

Alois P. Heinz, Table of n, a(n) for n = 0..10000

a(n) = 1/2*(A066183(n) - A066186(n)). - Vladeta Jovovic, Mar 23 2003

G.f.: Sum_{k>=1} x^(2*k)/(1 - x^k)^3 / Product_{j>=1} (1 - x^j). - Ilya Gutkovskiy, Mar 05 2021

a(n) = Sum_{k=0..A161680(n)} k * A264034(n,k). - Alois P. Heinz, Jan 20 2023

a(n) ~ 3 * zeta(3) * sqrt(n) * exp(Pi*sqrt(2*n/3)) / (sqrt(2) * Pi^3). - Vaclav Kotesovec, Jul 06 2025

a(3)=4 because the first moments of all partitions of 3 are {3}.{0},{2,1}.{0,1} and {1,1,1}.{0,1,2}, resulting in 0,1,3; summing to 4.

b:= proc(n, i) option remember; `if`(n=0 or i=1, [1, 0],

b(n, i-1)+(h-> h+[0, h[1]*i*(i-1)/2])(b(n-i, min(n-i, i))))

end:

a:= n-> b(n$2)[2]:

seq(a(n), n=0..50); # Alois P. Heinz, Jan 29 2014

Table[ Plus@@ Map[ #.Range[ 0, -1+Length[ # ] ]&, IntegerPartitions[ n ] ], {n, 40} ]

b[n_, i_] := b[n, i] = If[n==0, {1, 0}, If[i<1, {0, 0}, If[i>n, b[n, i-1], b[n, i-1] + Function[h, h+{0, h[[1]]*i*(i-1)/2}][b[n-i, i]]]]]; a[n_] := b[n, n][[2]]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Oct 26 2015, after Alois P. Heinz *)

Cf. A000337, A001788, A066183, A066184, A066186, A161680, A264034.

Sequence in context: A009844 A264100 A316540 * A330704 A239940 A320923

Adjacent sequences: A066182 A066183 A066184 * A066186 A066187 A066188

easy,nonn

Wouter Meeussen, Dec 15 2001

approved