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A240308
Number of partitions p of n such that (maximal multiplicity of the parts of p) >= (number of distinct parts of p).
4
0, 1, 2, 2, 4, 5, 8, 10, 15, 18, 28, 35, 48, 63, 85, 106, 141, 180, 229, 294, 374, 468, 591, 741, 925, 1149, 1421, 1751, 2163, 2648, 3239, 3944, 4813, 5825, 7062, 8518, 10286, 12340, 14835, 17739, 21223, 25287, 30155, 35787, 42522, 50296, 59556, 70243, 82902
OFFSET
0,3
FORMULA
a(n) = A239964(n) + A240308(n) for n >= 0.
a(n) + A240305(n) = A000041(n) for n >= 0.
EXAMPLE
a(6) counts these 8 partitions: 6, 411, 33, 3111, 222, 2211, 21111, 111111.
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 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Apr 05 2014
STATUS
approved