cp's OEIS Frontend

A239143 Number of strict partitions of n having standard deviation σ >= 1.

%I A239143 #5 Mar 14 2014 11:04:28
%S A239143 0,0,0,1,1,2,3,5,5,9,10,13,16,21,24,31,36,44,52,63,73,88,102,120,140,
%T A239143 164,189,221,254,294,338,389,445,511,583,666,758,863,979,1112,1258,
%U A239143 1424,1608,1815,2045,2303,2588,2908,3262,3657,4094,4581,5118,5716,6376
%N A239143 Number of strict partitions of n having standard deviation σ >= 1.
%C A239143 Regarding standard deviation, see Comments at A238616.
%F A239143 a(n) + A239140(n) = A000009(n) for n >=1.
%F A239143 G.f.: Product_{m>=1} (1+x^m) -1 +(x^6+x^5+x^4+2*x^3+3*x^2+2*x+1)*x / ((x-1)*(x+1)*(x^2+x+1)). - _Alois P. Heinz_, Mar 14 2014
%e A239143 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.
%t A239143 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]]
%t A239143 Table[Count[g[n], p_ /; s[p] < 1], {n, z}]   (* A239140 *)
%t A239143 Table[Count[g[n], p_ /; s[p] <= 1], {n, z}]  (* A239141 *)
%t A239143 Table[Count[g[n], p_ /; s[p] == 1], {n, z}]  (* periodic 01 *)
%t A239143 Table[Count[g[n], p_ /; s[p] > 1], {n, z}]   (* A239142 *)
%t A239143 Table[Count[g[n], p_ /; s[p] >= 1], {n, z}]  (* A239143 *)
%t A239143 t[n_] := t[n] = N[Table[s[g[n][[k]]], {k, 1, PartitionsQ[n]}]]
%t A239143 ListPlot[Sort[t[30]]] (*plot of st.dev's of strict partitions of 30*)
%t A239143 (* _Peter J. C. Moses_, Mar 03 2014 *)
%Y A239143 Cf. A239140, A239141, A239142, A000009, A238616.
%K A239143 nonn,easy
%O A239143 1,6
%A A239143 _Clark Kimberling_, Mar 11 2014