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A066187
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Sum of the second moments of all partitions of n.
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1
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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
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OFFSET
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0,3
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COMMENTS
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The first element of each partition is given weight 1.
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LINKS
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EXAMPLE
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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.
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MAPLE
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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]:
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MATHEMATICA
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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]];
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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STATUS
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approved
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