A240306 Number of partitions p of n such that (maximal multiplicity of the parts of p) <= (number of distinct parts of p).

1, 1, 1, 2, 3, 5, 6, 9, 13, 17, 25, 33, 44, 59, 76, 100, 131, 169, 212, 278, 352, 442, 555, 703, 871, 1088, 1342, 1664, 2046, 2517, 3064, 3758, 4574, 5548, 6718, 8119, 9797, 11784, 14150, 16935, 20263, 24179, 28798, 34237, 40677, 48122, 57008, 67291, 79400

Offset

0,4

Links

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Formula

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

a(n) + A240308(n) = A000041(n) for n >= 0.

G.f.: Sum_{i>=0} [z^i] Product_{j>=1} (1 + z * Sum_{k=1..i} q^(j*k)). - Seiichi Manyama, Mar 13 2026

Example

a(6) counts these 6 partitions:  6, 51, 42, 411, 321, 2211.

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 *)

Prog

(PARI) my(N=50, q='q+O('q^N)); Vec(sum(i=0, N, polcoef(prod(j=1, N, 1+z*sum(k=1, i, q^(j*k))), i, z))) \\ Seiichi Manyama, Mar 13 2026

Crossrefs

Cf. A239964, A240305, A239964, A240308, A240309, A000041, A394188.

Sequence in context: A330228 A166048 A396177 * A381992 A382077 A094873

Adjacent sequences: A240303 A240304 A240305 * A240307 A240308 A240309

Keyword

nonn,easy

Author

Clark Kimberling, Apr 05 2014

Status

approved