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a(n) = number of partitions p of n such that the greatest multiplicity of the parts of p is not a part of p.
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%I #8 Sep 22 2023 05:25:14

%S 1,0,2,2,2,5,7,11,13,19,24,37,47,65,84,112,141,190,235,308,388,498,

%T 617,789,973,1225,1508,1881,2298,2851,3467,4264,5172,6317,7620,9266,

%U 11132,13456,16117,19382,23127,27705,32940,39287,46579,55339,65397,77443,91248

%N a(n) = number of partitions p of n such that the greatest multiplicity of the parts of p is not a part of p.

%F a(n) = A000041(n) - A365613(n).

%e 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 do not include the greatest multiplicity as a part are [5], [3,2], [3,1,1], [2,1,1,1],and [1,1,1,1,1], so that a(5) = 5.

%t z = 40; f[n_] := f[n] = IntegerPartitions[n];

%t m[p_] := Max[Map[Length, Split[p]]]

%t Table[Count[f[n], p_ /; ! MemberQ[p, m[p]]], {n, 0, z}]

%Y Cf. A000041, A365613, A365614, A365615.

%K nonn

%O 0,3

%A _Clark Kimberling_, Sep 17 2023