A238006 Number of strict partitions of n such that (greatest part) - (least part) > (number of parts).

0, 0, 0, 0, 1, 1, 2, 3, 5, 6, 8, 11, 14, 18, 22, 27, 33, 41, 49, 59, 70, 83, 98, 116, 136, 159, 186, 215, 249, 289, 333, 383, 441, 505, 578, 660, 752, 856, 974, 1105, 1252, 1418, 1602, 1808, 2039, 2295, 2581, 2901, 3255, 3649, 4088, 4573, 5111, 5709, 6368

Offset: 1

Author

Clark Kimberling, Feb 17 2014

Keywords

• nonn
• easy

Examples

			a(8) = 3 counts these partitions:  7+1, 6+2, 5+2+1.
		

Crossrefs

Cf. A000009, A001227, A003056, A238005, A238007.

Programs

Mathematica

z = 70; q[n_] := q[n] = Select[IntegerPartitions[n], Max[Length /@ Split@#] == 1 &]; p1[p_] := p1[p] = DeleteDuplicates[p]; t[p_] := t[p] = Length[p1[p]];
Table[Count[q[n], p_ /; Max[p] - Min[p] < t[p]], {n, z}]  (* A001227 *)
Table[Count[q[n], p_ /; Max[p] - Min[p] <= t[p]], {n, z}] (* A003056 *)
Table[Count[q[n], p_ /; Max[p] - Min[p] == t[p]], {n, z}] (* A238005 *)
Table[Count[q[n], p_ /; Max[p] - Min[p] > t[p]], {n, z}]  (* A238006 *)
Table[Count[q[n], p_ /; Max[p] - Min[p] >= t[p]], {n, z}] (* A238007 *)

Formula

A001227(n) + A238005(n) + a(n) = A000009(n). - R. J. Mathar, Sep 08 2021

From Omar E. Pol, Sep 11 2021: (Start)

a(n) = A000009(n) - A003056(n).

a(n) = A238007(n) - A238005(n). (End)