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A238617
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Number of partitions of n having standard deviation σ <= 1.
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5
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1, 2, 3, 5, 6, 9, 10, 14, 15, 20, 23, 30, 33, 42, 44, 55, 58, 71, 78, 91, 99, 118, 124, 148, 155, 187, 202, 229, 244, 279, 290, 327, 344, 397, 427, 475, 501, 558, 597, 665, 714, 776, 824, 898, 948, 1032, 1084, 1245, 1308, 1395, 1452, 1606, 1692, 1807, 1919
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OFFSET
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1,2
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COMMENTS
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Regarding "standard deviation" see Comments at A238616.
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LINKS
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FORMULA
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EXAMPLE
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There are 11 partitions of 6, whose standard deviations are given by these approximations: 0., 2., 1., 1.41421, 0., 0.816497, 0.866025, 0., 0.5, 0.4, 0, so that a(6) = 9.
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MAPLE
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b:= proc(n, i, m, s, c) `if`(n=0, `if`(s/c-(m/c)^2<=1, 1, 0),
`if`(i=1, b(0$2, m+n, s+n, c+n), add(b(n-i*j, i-1,
m+i*j, s+i^2*j, c+j), j=0..n/i)))
end:
a:= n-> b(n$2, 0$3):
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MATHEMATICA
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z = 55; g[n_] := g[n] = IntegerPartitions[n]; s[t_] := s[t] = Sqrt[Sum[(t[[k]] - Mean[t])^2, {k, 1, Length[t]}]/Length[t]]
Table[Count[g[n], p_ /; s[p] < 1], {n, z}] (*A238616*)
Table[Count[g[n], p_ /; s[p] <= 1], {n, z}] (*A238617*)
Table[Count[g[n], p_ /; s[p] == 1], {n, z}] (*A238618*)
Table[Count[g[n], p_ /; s[p] > 1], {n, z}] (*A238619*)
Table[Count[g[n], p_ /; s[p] >= 1], {n, z}] (*A238620*)
t[n_] := t[n] = N[Table[s[g[n][[k]]], {k, 1, PartitionsP[n]}]]
ListPlot[Sort[t[30]]] (*plot of st.dev's of partitions of 30*)
b[n_, i_, m_, s_, c_] := b[n, i, m, s, c] = If[n == 0, If[s/c - (m/c)^2 <= 1, 1, 0], If[i == 1, b[0, 0, m + n, s + n, c + n], Sum[b[n - i*j, i - 1, m + i*j, s + i^2*j, c + j], {j, 0, n/i}]]]; a[n_] := b[n, n, 0, 0, 0]; Table[a[n], {n, 1, 50}] (* Jean-François Alcover, Nov 16 2015, after Alois P. Heinz *)
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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