A066184 Sum of the first moments of all partitions of n.

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

0,3

Comments

The first element of each partition is given weight 1.

Links

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

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

Example

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.

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 *)

Crossrefs

Cf. A066185.

Sequence in context: A046789 A271902 A272539 * A231799 A146924 A342032

Adjacent sequences: A066181 A066182 A066183 * A066185 A066186 A066187

Keyword

easy,nonn

Author

Wouter Meeussen, Dec 15 2001

Status

approved