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A238662
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Number of partitions of n having population standard deviation >= 2.
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6
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0, 0, 0, 0, 0, 1, 1, 3, 5, 9, 12, 20, 29, 43, 62, 88, 118, 169, 223, 306, 403, 532, 693, 907, 1160, 1490, 1910, 2423, 3044, 3845, 4783, 5957, 7401, 9104, 11209, 13805, 16806, 20449, 24920, 30223, 36494, 44022, 52880, 63511, 76003, 90631, 108088, 128708
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OFFSET
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1,8
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COMMENTS
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Regarding "population standard deviation" see Comments at A238616.
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LINKS
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Table of n, a(n) for n=1..48.
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FORMULA
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a(n) + A238658(n) = A000041(n).
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EXAMPLE
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There are 22 partitions of 8, whose population standard deviations are given by these approximations: 0., 3., 2., 2.35702, 1., 1.69967, 1.73205, 0., 1.24722, 0.942809, 1.22474, 1.2, 0.471405, 1., 0.707107, 0.8, 0.745356, 0., 0.489898, 0.471405, 0.349927, 0, so that a(8) = 3.
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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>=4, 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):
seq(a(n), n=1..50); # Alois P. Heinz, Mar 11 2014
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MATHEMATICA
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z = 50; g[n_] := g[n] = IntegerPartitions[n]; c[t_] := c[t] = Length[t]; s[t_] := s[t] = Sqrt[Sum[(t[[k]] - Mean[t])^2, {k, 1, c[t]}]/c[t]];
Table[Count[g[n], p_ /; s[p] < 2], {n, z}] (*A238658*)
Table[Count[g[n], p_ /; s[p] <= 2], {n, z}] (*A238659*)
Table[Count[g[n], p_ /; s[p] == 2], {n, z}] (*A238660*)
Table[Count[g[n], p_ /; s[p] > 2], {n, z}] (*A238661*)
Table[Count[g[n], p_ /; s[p] >= 2], {n, z}] (*A238662*)
t[n_] := t[n] = N[Table[s[g[n][[k]]], {k, 1, PartitionsP[n]}]]
ListPlot[Sort[t[30]]] (* plot of st deviations of partitions of 30 *)
(* Second program: *)
b[n_, i_, m_, s_, c_] := b[n, i, m, s, c] = If[n == 0, If[s/c - (m/c)^2 >= 4, 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];
Array[a, 50] (* Jean-François Alcover, May 27 2021, after Alois P. Heinz *)
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CROSSREFS
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Cf. A238616, A238658, A238660, A238661.
Sequence in context: A046746 A344715 A058599 * A059093 A084593 A275843
Adjacent sequences: A238659 A238660 A238661 * A238663 A238664 A238665
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KEYWORD
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nonn
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AUTHOR
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Clark Kimberling, Mar 03 2014
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STATUS
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approved
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