A066187 Sum of the second moments of all partitions of n with weights starting from 1.

0, 1, 7, 23, 66, 145, 321, 600, 1137, 1964, 3379, 5463, 8888, 13714, 21206, 31737, 47319, 68727, 99718, 141488, 200383, 279097, 387302, 530286, 724113, 976949, 1314106, 1751079, 2325412, 3062942, 4022617, 5244455, 6817630, 8808369, 11346219, 14536656, 18573495

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..1000

Example

a(3) = 23 because the second moments of all partitions of 3 are {3}.{1},{2,1}.{1,4} and {1,1,1}.{1,4,9}, resulting in 3,6,14; summing to 23.

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, 1)[2]:
seq(a(n), n=0..50);  # Alois P. Heinz, Jan 29 2014

Mathematica

Table[ Plus@@Map[ #.Range[ 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] +
     # + {0, #[[1]]*i*t^2}& @ b[n - i, i, t + 1]]]];
a[n_] := b[n, n, 1][[2]];
a /@ Range[0, 50] (* Jean-François Alcover, Jun 05 2021, after Alois P. Heinz *)

Crossrefs

Sequence in context: A343519 A365439 A266801 * A259214 A114246 A297315

Adjacent sequences: A066184 A066185 A066186 * A066188 A066189 A066190

Keyword

easy,nonn

Author

Wouter Meeussen, Dec 15 2001

Status

approved