cp's OEIS Frontend

A238617 Number of partitions of n having standard deviation σ <= 1.

%I A238617 #23 Nov 16 2015 14:59:15
%S A238617 1,2,3,5,6,9,10,14,15,20,23,30,33,42,44,55,58,71,78,91,99,118,124,148,
%T A238617 155,187,202,229,244,279,290,327,344,397,427,475,501,558,597,665,714,
%U A238617 776,824,898,948,1032,1084,1245,1308,1395,1452,1606,1692,1807,1919
%N A238617 Number of partitions of n having standard deviation σ <= 1.
%C A238617 Regarding "standard deviation" see Comments at A238616.
%F A238617 a(n) + A238619(n) = A000041(n).
%e A238617 There are 11 partitions of 6, whose standard deviations are given by these approximations:  0., 2., 1., 1.41421, 0., 0.816497, 0.866025, 0., 0.5, 0.4, 0, so that a(6) = 9.
%p A238617 b:= proc(n, i, m, s, c) `if`(n=0, `if`(s/c-(m/c)^2<=1, 1, 0),
%p A238617       `if`(i=1, b(0$2, m+n, s+n, c+n), add(b(n-i*j, i-1,
%p A238617        m+i*j, s+i^2*j, c+j), j=0..n/i)))
%p A238617     end:
%p A238617 a:= n-> b(n$2, 0$3):
%p A238617 seq(a(n), n=1..50);  # _Alois P. Heinz_, Mar 11 2014
%t A238617 z = 55; g[n_] := g[n] = IntegerPartitions[n]; s[t_] := s[t] = Sqrt[Sum[(t[[k]] - Mean[t])^2, {k, 1, Length[t]}]/Length[t]]
%t A238617 Table[Count[g[n], p_ /; s[p] < 1], {n, z}]   (*A238616*)
%t A238617 Table[Count[g[n], p_ /; s[p] <= 1], {n, z}]  (*A238617*)
%t A238617 Table[Count[g[n], p_ /; s[p] == 1], {n, z}]  (*A238618*)
%t A238617 Table[Count[g[n], p_ /; s[p] > 1], {n, z}]   (*A238619*)
%t A238617 Table[Count[g[n], p_ /; s[p] >= 1], {n, z}]  (*A238620*)
%t A238617 t[n_] := t[n] = N[Table[s[g[n][[k]]], {k, 1, PartitionsP[n]}]]
%t A238617 ListPlot[Sort[t[30]]] (*plot of st.dev's of partitions of 30*)
%t A238617 b[n_, i_, m_, s_, c_] := b[n, i, m, s, c] = If[n == 0, If[s/c - (m/c)^2 <= 1, 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]; Table[a[n], {n, 1, 50}] (* _Jean-François Alcover_, Nov 16 2015, after _Alois P. Heinz_ *)
%Y A238617 Cf. A238616.
%K A238617 nonn,easy
%O A238617 1,2
%A A238617 _Clark Kimberling_, Mar 01 2014