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A066187 Sum of the second moments of all partitions of n. 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 (list; graph; refs; listen; history; text; internal format)
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-Fran├žois Alcover, Jun 05 2021, after Alois P. Heinz *)

CROSSREFS

Sequence in context: A306971 A343519 A266801 * A259214 A114246 A297315

Adjacent sequences:  A066184 A066185 A066186 * A066188 A066189 A066190

KEYWORD

easy,nonn

AUTHOR

Wouter Meeussen, Dec 15 2001

STATUS

approved

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Last modified January 21 12:58 EST 2022. Contains 350477 sequences. (Running on oeis4.)