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

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, 164, 189, 221, 254, 294, 338, 389, 445, 511, 583, 666, 758, 863, 979, 1112, 1258, 1424, 1608, 1815, 2045, 2303, 2588, 2908, 3262, 3657, 4094, 4581, 5118, 5716, 6376

Offset: 1

Author

Clark Kimberling, Mar 11 2014

Keywords

• nonn
• easy

Comments

Regarding standard deviation, see Comments at A238616.

Examples

			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.
		

Crossrefs

Cf. A239140, A239141, A239142, A000009, A238616.

Programs

Mathematica

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]]
Table[Count[g[n], p_ /; s[p] < 1], {n, z}]   (* A239140 *)
Table[Count[g[n], p_ /; s[p] <= 1], {n, z}]  (* A239141 *)
Table[Count[g[n], p_ /; s[p] == 1], {n, z}]  (* periodic 01 *)
Table[Count[g[n], p_ /; s[p] > 1], {n, z}]   (* A239142 *)
Table[Count[g[n], p_ /; s[p] >= 1], {n, z}]  (* A239143 *)
t[n_] := t[n] = N[Table[s[g[n][[k]]], {k, 1, PartitionsQ[n]}]]
ListPlot[Sort[t[30]]] (*plot of st.dev's of strict partitions of 30*)
(* Peter J. C. Moses, Mar 03 2014 *)

Formula

a(n) + A239140(n) = A000009(n) for n >=1.

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