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A365613
a(n) = number of partitions p of n such that the greatest multiplicity of the parts of p is a part of p.
3
0, 1, 0, 1, 3, 2, 4, 4, 9, 11, 18, 19, 30, 36, 51, 64, 90, 107, 150, 182, 239, 294, 385, 466, 602, 733, 928, 1129, 1420, 1714, 2137, 2578, 3177, 3826, 4690, 5617, 6845, 8181, 9898, 11803, 14211, 16878, 20234, 23974, 28596, 33795, 40161, 47311, 56025, 65845
OFFSET
0,5
FORMULA
a(n) = A000041(n) - A365616(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 greatest multiplicities 1,1,1,2,2,3,5, respectively. The partitions that include greatest multiplicity as a part are [4,1] and [2,2,1], so that a(5) = 2.
MATHEMATICA
z = 40; f[n_] := f[n] = IntegerPartitions[n];
m[p_] := Max[Map[Length, Split[p]]]
Table[Count[f[n], p_ /; MemberQ[p, m[p]]], {n, 0, z}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Clark Kimberling, Sep 17 2023
STATUS
approved