A241822 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 distinct parts of p.

0, 0, 0, 0, 0, 1, 2, 4, 5, 10, 12, 19, 27, 37, 51, 70, 90, 119, 161, 200, 259, 335, 428, 533, 681, 845, 1065, 1317, 1636, 2010, 2496, 3036, 3719, 4538, 5523, 6686, 8096, 9760, 11759, 14138, 16943, 20263, 24216, 28852, 34308, 40801, 48315, 57206, 67590, 79748

Offset

0,7

Links

Table of n, a(n) for n=0..49.

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]]; g[p_] := Max[-Differences[p]];
Table[Count[f[n], p_ /; g[p] < d[p]], {n, 0, z}]  (* A241818 *)
Table[Count[f[n], p_ /; g[p] <= d[p]], {n, 0, z}] (* A241819 *)
Table[Count[f[n], p_ /; g[p] == d[p]], {n, 0, z}] (* A241820 *)
Table[Count[f[n], p_ /; g[p] >= d[p]], {n, 0, z}] (* A241821 *)
Table[Count[f[n], p_ /; g[p] > d[p]], {n, 0, z}]  (* A241822 *)

Crossrefs

Cf. A241818, A241819, A241820, A241821, A000041.

Sequence in context: A355149 A022944 A370593 * A133732 A328221 A364913

Adjacent sequences: A241819 A241820 A241821 * A241823 A241824 A241825

Keyword

nonn,easy

Author

Clark Kimberling, Apr 30 2014

Status

approved