A239954 Number of partitions p of n such that (number of distinct parts of p) < max(p) - min(p).

0, 0, 0, 0, 0, 1, 2, 4, 6, 12, 17, 26, 38, 54, 76, 107, 142, 192, 259, 337, 443, 577, 743, 948, 1213, 1532, 1935, 2427, 3031, 3765, 4681, 5762, 7097, 8704, 10644, 12966, 15775, 19104, 23115, 27874, 33546, 40257, 48259, 57656, 68809, 81929, 97378, 115495

Offset: 0

Author

Clark Kimberling, Mar 30 2014

Keywords

• nonn
• easy

Examples

			a(7) counts these 4 partitions:  61, 52, 511, 1111111.
		

Crossrefs

Cf. A239955, A239956, A034296, A239958.

Programs

Mathematica

z = 60; d[p_] := d[p] = Length[DeleteDuplicates[p]]; f[p_] := f[p] = Max[p] - Min[p]; g[n_] := g[n] = IntegerPartitions[n];
Table[Count[g[n], p_ /; d[p] < f[p]], {n, 0, z}]  (*A239954*)
Table[Count[g[n], p_ /; d[p] <= f[p]], {n, 0, z}] (*A239955*)
Table[Count[g[n], p_ /; d[p] == f[p]], {n, 0, z}] (*A239956*)
Table[Count[g[n], p_ /; d[p] > f[p]], {n, 0, z}]  (*A034296*)
Table[Count[g[n], p_ /; d[p] >= f[p]], {n, 0, z}] (*A239958*)

Formula

a(n) + A239958(n) = A000041(n) for n >= 0.