A241831 Number of partitions p = [x(1), ..., x(k)], where x(1) >= x(2) >= ... >= x(k), of n such that max(x(i) - x(i-1)) >= number of parts of p.

0, 0, 0, 0, 1, 1, 3, 3, 6, 8, 12, 15, 23, 27, 38, 48, 64, 78, 105, 127, 165, 202, 256, 311, 393, 473, 588, 711, 875, 1050, 1286, 1537, 1867, 2229, 2687, 3195, 3838, 4544, 5427, 6416, 7625, 8981, 10637, 12492, 14736, 17269, 20293, 23715, 27792, 32391, 37840

Offset

0,7

Links

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

Formula

a(n) = A241830(n) + A241832(n).

a(n) + A241828(n) = A000041(n) for n >= 0.

Example

a(6) counts these 3 partitions:  51, 42, 411.

Mathematica

z = 30; f[n_] := f[n] = IntegerPartitions[n]; g[p_] := Max[-Differences[p]]
Table[Count[f[n], p_ /; g[p] < Length[p]], {n, 0, z}]  (* A241828 *)
Table[Count[f[n], p_ /; g[p] <= Length[p]], {n, 0, z}] (* A241829 *)
Table[Count[f[n], p_ /; g[p] == Length[p]], {n, 0, z}] (* A241830 *)
Table[Count[f[n], p_ /; g[p] >= Length[p]], {n, 0, z}] (* A241831 *)
Table[Count[f[n], p_ /; g[p] > Length[p]], {n, 0, z}]  (* A241832 *)

Crossrefs

Cf. A241828, A241829, A241830, A241832, A000041.

Sequence in context: A309455 A168637 A241390 * A239946 A130780 A174524

Adjacent sequences: A241828 A241829 A241830 * A241832 A241833 A241834

Keyword

nonn,easy

Author

Clark Kimberling, Apr 30 2014

Status

approved