A241823 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, 3, 5, 8, 12, 18, 26, 37, 51, 70, 94, 126, 166, 219, 284, 369, 473, 607, 770, 977, 1228, 1544, 1925, 2399, 2970, 3673, 4517, 5550, 6784, 8284, 10073, 12232, 14799, 17883, 21536, 25903, 31064, 37204, 44439, 53015, 63090, 74987, 88932, 105337

Offset

0,4

Links

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

Formula

a(n) = A241821(n) - A241820(n) for n >= 0.

a(n) + A241818(n) + A241820(n) = A000041(n) for n >= 0.

Example

a(6) counts these 2 partitions:  51, 411

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. A241824, A241825, A241826, A241827, A000041.

Sequence in context: A039901 A173564 A121946 * A058984 A084376 A098693

Adjacent sequences: A241820 A241821 A241822 * A241824 A241825 A241826

Keyword

nonn,easy

Author

Clark Kimberling, Apr 30 2014

Status

approved