A240309 Number of partitions p of n such that (maximal multiplicity of the parts of p) > (number of distinct parts of p).

0, 0, 1, 1, 2, 2, 5, 6, 9, 13, 17, 23, 33, 42, 59, 76, 100, 128, 173, 212, 275, 350, 447, 552, 704, 870, 1094, 1346, 1672, 2048, 2540, 3084, 3775, 4595, 5592, 6764, 8180, 9853, 11865, 14250, 17075, 20404, 24376, 29024, 34498, 41012, 48550, 57463, 67873

Offset

0,5

Links

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

Formula

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

a(n) + A240305(n) + A239964(n) = A000041(n) for n >= 0.

Example

a(6) counts these 5 partitions:  33, 3111, 222, 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, A240308, A000041.

Sequence in context: A258619 A348757 A118807 * A098507 A097066 A035548

Adjacent sequences: A240306 A240307 A240308 * A240310 A240311 A240312

Keyword

nonn,easy

Author

Clark Kimberling, Apr 05 2014

Status

approved