A240305 Number of partitions p of n such that (maximal multiplicity of the parts of p) < (number of distinct parts of p).

1, 0, 0, 1, 1, 2, 3, 5, 7, 12, 14, 21, 29, 38, 50, 70, 90, 117, 156, 196, 253, 324, 411, 514, 650, 809, 1015, 1259, 1555, 1917, 2365, 2898, 3536, 4318, 5248, 6365, 7691, 9297, 11180, 13446, 16115, 19296, 23019, 27474, 32653, 38838, 46002, 54511, 64371, 76012

Offset

0,6

Links

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

Formula

a(n) = A240306(n) - A239964(n) for n >= 0.

a(n) + A239964(n) + A240309(n) = A000041(n) for n >= 0.

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

Example

a(6) counts these 3 partitions:  51, 42, 321.

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-1, q^(j*k))), i, z))) \\ Seiichi Manyama, Mar 13 2026

Crossrefs

Cf. A240306, A239964, A240308, A240309, A000041, A394189.

Sequence in context: A031216 A004683 A363077 * A100036 A179781 A309370

Adjacent sequences: A240302 A240303 A240304 * A240306 A240307 A240308

Keyword

nonn,easy

Author

Clark Kimberling, Apr 05 2014

Status

approved