A365614 a(n) = number of partitions p of n such that the least multiplicity of the parts of p is a part of p.

0, 1, 0, 1, 3, 4, 6, 10, 13, 20, 27, 36, 52, 71, 94, 126, 170, 216, 286, 367, 473, 603, 771, 963, 1229, 1529, 1910, 2371, 2959, 3623, 4492, 5487, 6740, 8200, 10016, 12099, 14724, 17722, 21402, 25687, 30914, 36892, 44224, 52630, 62781, 74497, 88540, 104646

Offset

0,5

Links

Table of n, a(n) for n=0..47.

Formula

a(n) = A000041(n) - A365615(n).

Example

The partitions of 5 are [5], [4,1], [3,2], [3,1,1], [2,2,1], [2,1,1,1], [1,1,1,1,1], having least multiplicities 1,1,1,1,1,1,5, respectively. The partitions that include least multiplicity as a part are [4,1], [3,1,1], [2,2,1], and [2,1,1,1], so that a(5) = 4.

Mathematica

z = 40; f[n_] := f[n] = IntegerPartitions[n];
m[p_] := Min[Map[Length, Split[p]]]
Table[Count[f[n], p_ /; MemberQ[p, m[p]]], {n, 0, z}]

Crossrefs

Cf. A000041, A365613, A365615, A365616.

Sequence in context: A310001 A310002 A240577 * A074321 A253049 A167410

Adjacent sequences: A365611 A365612 A365613 * A365615 A365616 A365617

Keyword

nonn

Author

Clark Kimberling, Sep 17 2023

Status

approved