A066188 Sum of the second moments of all partitions of n.

0, 0, 1, 6, 22, 58, 141, 289, 579, 1054, 1885, 3161, 5280, 8371, 13220, 20183, 30605, 45178, 66448, 95546, 136877, 192759, 270146, 373387, 514187, 699429, 947846, 1272067, 1700456, 2254007, 2977623, 3904091, 5102058, 6625434, 8575277, 11037277, 14163777

Offset

0,4

Comments

The first element of each partition is given weight 0.

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000 (terms 0..1000 from Alois P. Heinz)

Formula

a(n) ~ exp(Pi*sqrt(2*n/3)) * n / (5*sqrt(3)). - Vaclav Kotesovec, May 30 2021

Example

a(3) = 6 because the second moments of all partitions of 3 are {3}.{0},{2,1}.{0,1} and {1,1,1}.{0,1,4}, resulting in 0,1,5; summing to 6.

Maple

b:= proc(n, i, t) option remember; `if`(n=0, [1, 0],
      `if`(i<1, [0$2], `if`(i>n, b(n, i-1, t), b(n, i-1, t)+
       (h-> h+[0, h[1]*i*t^2])(b(n-i, i, t+1)))))
    end:
a:= n-> b(n$2, 0)[2]:
seq(a(n), n=0..50);  # Alois P. Heinz, Jan 29 2014

Mathematica

Table[ Plus@@Map[ #.Range[ 0, -1+Length[ # ] ]^2&, IntegerPartitions[ n ] ], {n, 30} ]
(* Second program: *)
b[n_, i_, t_] := b[n, i, t] = If[n == 0, {1, 0},
     If[i < 1, {0, 0}, If[i > n, b[n, i - 1, t], b[n, i - 1, t] +
     Function[h, h + {0, h[[1]]*i*t^2}][b[n - i, i, t + 1]]]]];
a[n_] := b[n, n, 0][[2]];
a /@ Range[0, 50] (* Jean-François Alcover, May 30 2021, after Alois P. Heinz *)

Crossrefs

Sequence in context: A366448 A127760 A320243 * A071239 A105450 A011888

Adjacent sequences: A066185 A066186 A066187 * A066189 A066190 A066191

Keyword

easy,nonn

Author

Wouter Meeussen, Dec 15 2001

Status

approved