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A238660 Number of partitions of n having population standard deviation = 2.

%I A238660 #22 Jun 03 2021 09:31:25
%S A238660 0,0,0,0,0,1,0,1,0,2,0,2,0,1,1,3,0,5,0,7,4,2,0,19,3,2,9,20,0,38,0,22,
%T A238660 33,7,12,84,0,8,52,90,0,127,0,87,103,22,0,304,9,74,131,153,0,214,139,
%U A238660 390,192,59,0,1219,0,73,460,372,383,908,0,501,439,832,0
%N A238660 Number of partitions of n having population standard deviation = 2.
%C A238660 Regarding "standard deviation" see Comments at A238616.
%e A238660 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) = 1.
%p A238660 b:= proc(n, i, m, s, c) `if`(n=0, `if`(s/c-(m/c)^2=4, 1, 0),
%p A238660       `if`(i=1, b(0$2, m+n, s+n, c+n), add(b(n-i*j, i-1,
%p A238660        m+i*j, s+i^2*j, c+j), j=0..n/i)))
%p A238660     end:
%p A238660 a:= n-> b(n$2, 0$3):
%p A238660 seq(a(n), n=1..50);  # _Alois P. Heinz_, Mar 11 2014
%t A238660 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 A238660 Table[Count[g[n], p_ /; s[p] < 2], {n, z}]   (*A238658*)
%t A238660 Table[Count[g[n], p_ /; s[p] <= 2], {n, z}]  (*A238659*)
%t A238660 Table[Count[g[n], p_ /; s[p] == 2], {n, z}]  (*A238660*)
%t A238660 Table[Count[g[n], p_ /; s[p] > 2], {n, z}]   (*A238661*)
%t A238660 Table[Count[g[n], p_ /; s[p] >= 2], {n, z}]  (*A238662*)
%t A238660 t[n_] := t[n] = N[Table[s[g[n][[k]]], {k, 1, PartitionsP[n]}]]
%t A238660 ListPlot[Sort[t[30]]] (*plot of st deviations of partitions of 30*)
%t A238660 (* Second program: *)
%t A238660 b[n_, i_, m_, s_, c_] := b[n, i, m, s, c] = If[n == 0,
%t A238660      If[s/c - (m/c)^2 == 4, 1, 0], If[i == 1, b[0, 0, m+n, s+n, c+n],
%t A238660      Sum[b[n - i*j, i - 1, m + i*j, s + i^2*j, c + j], {j, 0, n/i}]]];
%t A238660 a[n_] := b[n, n, 0, 0, 0];
%t A238660 Array[a, 50] (* _Jean-François Alcover_, Jun 03 2021, after _Alois P. Heinz_ *)
%Y A238660 Cf. A238616, A238658-A238660, A238662.
%K A238660 nonn,easy
%O A238660 1,10
%A A238660 _Clark Kimberling_, Mar 03 2014
%E A238660 a(51)-a(71) from _Alois P. Heinz_, Mar 11 2014