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A238659 Number of partitions of n having standard deviation σ <= 2.

%I A238659 #15 Mar 11 2014 13:39:56
%S A238659 1,2,3,5,7,11,14,20,25,35,44,59,72,93,115,146,179,221,267,328,393,472,
%T A238659 562,687,801,948,1109,1315,1521,1797,2059,2414,2775,3213,3686,4256,
%U A238659 4831,5574,6317,7205,8089,9279,10381,11751,13234,14949,16666,18869,20986
%N A238659 Number of partitions of n having standard deviation σ <= 2.
%C A238659 Regarding "standard deviation" see Comments at A238616.
%F A238659 a(n) + A238661(n) = A000041(n).
%e A238659 There are 22 partitions of 8, whose 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) = 20.
%t A238659 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]];
%t A238659 Table[Count[g[n], p_ /; s[p] < 2], {n, z}]   (*A238658*)
%t A238659 Table[Count[g[n], p_ /; s[p] <= 2], {n, z}]  (*A238659*)
%t A238659 Table[Count[g[n], p_ /; s[p] == 2], {n, z}]  (*A238660*)
%t A238659 Table[Count[g[n], p_ /; s[p] > 2], {n, z}]   (*A238661*)
%t A238659 Table[Count[g[n], p_ /; s[p] >= 2], {n, z}]  (*A238662*)
%t A238659 t[n_] := t[n] = N[Table[s[g[n][[k]]], {k, 1, PartitionsP[n]}]]
%t A238659 ListPlot[Sort[t[30]]] (*plot of st deviations of partitions of 30*)
%Y A238659 Cf. A238616, A238658, A238660-A238662.
%K A238659 nonn,easy
%O A238659 1,2
%A A238659 _Clark Kimberling_, Mar 03 2014