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A240312
Number of partitions p of n such that (maximal multiplicity of the parts of p) = (maximal part of p).
5
1, 1, 0, 0, 2, 1, 2, 0, 2, 3, 5, 5, 9, 7, 11, 11, 18, 15, 28, 27, 41, 43, 62, 64, 91, 96, 127, 140, 184, 200, 260, 287, 365, 410, 511, 573, 717, 803, 985, 1120, 1359, 1538, 1859, 2106, 2522, 2870, 3407, 3872, 4586, 5207, 6128, 6976, 8167, 9284, 10844, 12321
OFFSET
0,5
FORMULA
a(n) = A240311(n) - A240310(n) for n >= 0.
a(n) + A240310(n) + A240314(n) = A000041(n) for n >= 0.
EXAMPLE
a(6) counts these 2 partitions: 3111, 2211.
MATHEMATICA
z = 60; f[n_] := f[n] = IntegerPartitions[n]; m[p_] := Max[Map[Length, Split[p]]] (* maximal multiplicity *)
Table[Count[f[n], p_ /; m[p] < Max[p]], {n, 0, z}] (* A240310 *)
Table[Count[f[n], p_ /; m[p] <= Max[p]], {n, 0, z}] (* A240311 *)
Table[Count[f[n], p_ /; m[p] == Max[p]], {n, 0, z}] (* A240312 *)
Table[Count[f[n], p_ /; m[p] >= Max[p]], {n, 0, z}] (* A240313 *)
Table[Count[f[n], p_ /; m[p] > Max[p]], {n, 0, z}] (* A240314 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Apr 05 2014
STATUS
approved