A241824 Number of partitions p = [x(1), ..., x(k)], where x(1) >= x(2) >= ... >= x(k), of n such that min(x(i) - x(i-1)) <= number of distinct parts of p.

0, 0, 1, 2, 4, 5, 9, 12, 19, 26, 38, 51, 72, 95, 129, 168, 223, 287, 374, 477, 613, 775, 984, 1234, 1552, 1932, 2408, 2978, 3684, 4527, 5563, 6797, 8301, 10090, 12253, 14821, 17910, 21564, 25936, 31099, 37245, 44482, 53064, 63142, 75046, 88994, 105406

Offset

0,4

Links

Table of n, a(n) for n=0..46.

Formula

a(n) = A241823(n) + A241824(n) for n >= 0.

a(n) + A241827(n) = A000041(n) for n >= 0.

Example

a(6) counts these 9 partitions:  42, 411, 33, 321, 3111, 222, 2211, 21111, 111111.

Mathematica

z = 30; f[n_] := f[n] = IntegerPartitions[n]; d[p_] := d[p] = Length[DeleteDuplicates[p]]; g1[p_] := Min[-Differences[p]]
Table[Count[f[n], p_ /; g1[p] < d[p]], {n, 0, z}]  (* A241823 *)
Table[Count[f[n], p_ /; g1[p] <= d[p]], {n, 0, z}] (* A241824 *)
Table[Count[f[n], p_ /; g1[p] == d[p]], {n, 0, z}] (* A241825 *)
Table[Count[f[n], p_ /; g1[p] >= d[p]], {n, 0, z}] (* A241826 *)
Table[Count[f[n], p_ /; g1[p] > d[p]], {n, 0, z}]  (* A241827 *)

Crossrefs

Cf. A241823, A241825, A241826, A241827, A000041.

Sequence in context: A371171 A039898 A083690 * A144121 A240576 A060312

Adjacent sequences: A241821 A241822 A241823 * A241825 A241826 A241827

Keyword

nonn,easy

Author

Clark Kimberling, Apr 30 2014

Status

approved