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A239142
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Number of strict partitions of n having standard deviation sigma > 1.
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4
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0, 0, 0, 0, 1, 1, 3, 4, 5, 8, 10, 12, 16, 20, 24, 30, 36, 43, 52, 62, 73, 87, 102, 119, 140, 163, 189, 220, 254, 293, 338, 388, 445, 510, 583, 665, 758, 862, 979, 1111, 1258, 1423, 1608, 1814, 2045, 2302, 2588, 2907, 3262, 3656, 4094, 4580, 5118, 5715, 6376
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OFFSET
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1,7
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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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G.f.: Product_{m>=1} (1+x^m) -1 +(x^5+x^4+x^3+2*x^2+x+1)*x / ((x-1)*(x^2+x+1)). - Alois P. Heinz, Mar 14 2014
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EXAMPLE
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The standard deviations of the strict partitions of 9 are 0., 3.5, 2.5, 1.5, 2.16025, 0.5, 1.63299, 0.816497, so that a(9) = 5.
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MATHEMATICA
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z = 30; g[n_] := Select[IntegerPartitions[n], Max[Length /@ Split@#] == 1 &]; 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}] (* A239140 *)
Table[Count[g[n], p_ /; s[p] <= 1], {n, z}] (* A239141 *)
Table[Count[g[n], p_ /; s[p] == 1], {n, z}] (* periodic 01 *)
Table[Count[g[n], p_ /; s[p] > 1], {n, z}] (* A239142 *)
Table[Count[g[n], p_ /; s[p] >= 1], {n, z}] (* A239143 *)
t[n_] := t[n] = N[Table[s[g[n][[k]]], {k, 1, PartitionsQ[n]}]]
ListPlot[Sort[t[30]]] (*plot of st.dev's of strict partitions of 30*)
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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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