A066184 Sum of the first moments of all partitions of n with weight starting at 1.

0, 1, 5, 13, 32, 61, 123, 208, 367, 590, 957, 1459, 2266, 3328, 4938, 7097, 10205, 14299, 20100, 27626, 38023, 51485, 69600, 92882, 123863, 163235, 214798, 280141, 364530, 470660, 606557, 776233, 991370, 1258827, 1594741, 2010142, 2528445, 3165648, 3955190

Offset: 0

Author

Wouter Meeussen, Dec 15 2001

Keywords

• easy
• nonn

Comments

The first element of each partition is given weight 1.

Examples

			a(3)=13 because the first moments of all partitions of 3 are {3}.{1},{2,1}.{1,2} and {1,1,1}.{1,2,3}, resulting in 3,4,6; summing to 13.
		

Links

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

Crossrefs

Cf. A066185.

Programs

Maple

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

Mathematica

Table[ Plus@@ Map[ #.Range[ Length[ # ] ]&, IntegerPartitions[ n ] ], {n, 40} ]
(* Second program: *)
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, Aug 29 2016, after Alois P. Heinz *)

Formula

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

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

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