A240308 Number of partitions p of n such that (maximal multiplicity of the parts of p) >= (number of distinct parts of p).

0, 1, 2, 2, 4, 5, 8, 10, 15, 18, 28, 35, 48, 63, 85, 106, 141, 180, 229, 294, 374, 468, 591, 741, 925, 1149, 1421, 1751, 2163, 2648, 3239, 3944, 4813, 5825, 7062, 8518, 10286, 12340, 14835, 17739, 21223, 25287, 30155, 35787, 42522, 50296, 59556, 70243, 82902

Offset

0,3

Links

Table of n, a(n) for n=0..48.

Formula

a(n) = A239964(n) + A240308(n) for n >= 0.

a(n) + A240305(n) = A000041(n) for n >= 0.

Example

a(6) counts these 8 partitions:  6, 411, 33, 3111, 222, 2211, 21111, 111111.

Mathematica

z = 60; f[n_] := f[n] = IntegerPartitions[n]; m[p_] := Max[Map[Length, Split[p]]]  (* maximal multiplicity *); d[p_] := d[p] = Length[DeleteDuplicates[p]] (* number of distinct terms *)
t1 = Table[Count[f[n], p_ /; m[p] < d[p]], {n, 0, z}]  (* A240305 *)
t2 = Table[Count[f[n], p_ /; m[p] <= d[p]], {n, 0, z}] (* A240306 *)
t3 = Table[Count[f[n], p_ /; m[p] == d[p]], {n, 0, z}] (* A239964 *)
t4 = Table[Count[f[n], p_ /; m[p] >= d[p]], {n, 0, z}] (* A240308 *)
t5 = Table[Count[f[n], p_ /; m[p] > d[p]], {n, 0, z}]  (* A240309 *)

Crossrefs

Cf. A240305, A240306, A239964, A240309, A000041.

Sequence in context: A323530 A271593 A131945 * A326526 A035951 A035957

Adjacent sequences: A240305 A240306 A240307 * A240309 A240310 A240311

Keyword

nonn,easy

Author

Clark Kimberling, Apr 05 2014

Status

approved