A240313 Number of partitions p of n such that (maximal multiplicity of the parts of p) >= (maximal part of p).

1, 1, 1, 1, 3, 3, 5, 5, 8, 11, 15, 19, 27, 32, 43, 53, 70, 84, 112, 135, 174, 212, 268, 324, 407, 490, 606, 731, 897, 1075, 1312, 1567, 1899, 2265, 2726, 3238, 3886, 4598, 5486, 6482, 7698, 9063, 10727, 12592, 14846, 17391, 20427, 23862, 27952, 32568, 38033

Offset

0,5

Links

Table of n, a(n) for n=0..50.

Formula

a(n) = A240312(n) + A240314(n) for n >= 0.

a(n) + A240310(n) = A000041(n) for n >= 0.

Example

a(6) counts these 5 partitions:  3111, 222, 2211, 21111, 111111.

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

Cf. A240310, A240311, A240312, A240314, A000041, A118053.

Sequence in context: A362059 A117772 A026924 * A275369 A062130 A151970

Adjacent sequences: A240310 A240311 A240312 * A240314 A240315 A240316

Keyword

nonn,easy

Author

Clark Kimberling, Apr 05 2014

Status

approved