cp's OEIS Frontend

A238662 Number of partitions of n having population standard deviation >= 2.

%I A238662 #29 May 29 2021 05:50:43
%S A238662 0,0,0,0,0,1,1,3,5,9,12,20,29,43,62,88,118,169,223,306,403,532,693,
%T A238662 907,1160,1490,1910,2423,3044,3845,4783,5957,7401,9104,11209,13805,
%U A238662 16806,20449,24920,30223,36494,44022,52880,63511,76003,90631,108088,128708
%N A238662 Number of partitions of n having population standard deviation >= 2.
%C A238662 Regarding "population standard deviation" see Comments at A238616.
%F A238662 a(n) + A238658(n) = A000041(n).
%e A238662 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.
%p A238662 b:= proc(n, i, m, s, c) `if`(n=0, `if`(s/c-(m/c)^2>=4, 1, 0),
%p A238662       `if`(i=1, b(0$2, m+n, s+n, c+n), add(b(n-i*j, i-1,
%p A238662        m+i*j, s+i^2*j, c+j), j=0..n/i)))
%p A238662     end:
%p A238662 a:= n-> b(n$2, 0$3):
%p A238662 seq(a(n), n=1..50);  # _Alois P. Heinz_, Mar 11 2014
%t A238662 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 A238662 Table[Count[g[n], p_ /; s[p] < 2], {n, z}]   (*A238658*)
%t A238662 Table[Count[g[n], p_ /; s[p] <= 2], {n, z}]  (*A238659*)
%t A238662 Table[Count[g[n], p_ /; s[p] == 2], {n, z}]  (*A238660*)
%t A238662 Table[Count[g[n], p_ /; s[p] > 2], {n, z}]   (*A238661*)
%t A238662 Table[Count[g[n], p_ /; s[p] >= 2], {n, z}]  (*A238662*)
%t A238662 t[n_] := t[n] = N[Table[s[g[n][[k]]], {k, 1, PartitionsP[n]}]]
%t A238662 ListPlot[Sort[t[30]]] (* plot of st deviations of partitions of 30 *)
%t A238662 (* Second program: *)
%t A238662 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}]]];
%t A238662 a[n_] := b[n, n, 0, 0, 0];
%t A238662 Array[a, 50] (* _Jean-François Alcover_, May 27 2021, after _Alois P. Heinz_ *)
%Y A238662 Cf. A238616, A238658, A238660, A238661.
%K A238662 nonn
%O A238662 1,8
%A A238662 _Clark Kimberling_, Mar 03 2014