A241064 Number of partitions p of n into distinct parts such that max(p) > 1 + 2*min(p).

0, 0, 0, 0, 0, 1, 1, 2, 4, 4, 6, 8, 11, 13, 16, 21, 26, 32, 38, 45, 56, 66, 79, 94, 110, 128, 151, 178, 207, 240, 277, 320, 370, 426, 488, 561, 642, 732, 834, 948, 1079, 1225, 1388, 1570, 1774, 2002, 2254, 2540, 2856, 3206, 3598, 4034, 4516, 5050, 5642, 6298

Offset: 0

Author

Clark Kimberling, Apr 16 2014

Keywords

• nonn
• easy

Examples

			a(10) counts these 6 partitions:  91, 82, 721, 631, 541, 4321
		

Crossrefs

Cf. A207642, A241061, A241062, A241064, A000009.

Programs

Mathematica

z = 70; f[n_] := f[n] = Select[IntegerPartitions[n], Max[Length /@ Split@#] == 1 &];
Table[Count[f[n], p_ /; Max[p] < 1 + 2*Min[p]], {n, 0, z}]  (* A241061 *)
Table[Count[f[n], p_ /; Max[p] <= 1 + 2*Min[p]], {n, 0, z}] (* A207642 *)
Table[Count[f[n], p_ /; Max[p] == 1 + 2*Min[p]], {n, 0, z}] (* A241062 *)
Table[Count[f[n], p_ /; Max[p] >= 1 + 2*Min[p]], {n, 0, z}] (* A241037 *)
Table[Count[f[n], p_ /; Max[p] > 1 + 2*Min[p]], {n, 0, z}]  (* A241064 *)
Table[Count[IntegerPartitions[n],?(Length[#]==Length[Union[#]]&&#[[1]]>2#[[-1]]+1&)],{n,0,60}]//Quiet (* _Harvey P. Dale, Sep 25 2024 *)

Formula

a(n) + A241061(n) + A241062(n) = A000009(n) for n >= 1.

a(n) = A241037(n) - A241062(n) for n>= 0.