A241820 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, 1, 1, 2, 2, 5, 3, 10, 10, 13, 18, 25, 25, 39, 48, 54, 78, 95, 113, 142, 183, 215, 270, 322, 396, 480, 587, 686, 845, 1022, 1210, 1453, 1730, 2081, 2459, 2945, 3454, 4108, 4838, 5744, 6707, 7959, 9216, 10938, 12692, 14934, 17346, 20296, 23526

Offset

0,7

Links

Table of n, a(n) for n=0..51.

Formula

a(n) = A241819(n) - A241818(n) for n >= 0.

a(n) + A241818(n) + A241822 (n) = A000041(n) for n >= 0.

Example

a(6) counts these 2 partitions:  42, 3111.

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, A241821, A241822, A000041.

Sequence in context: A327754 A308096 A006800 * A190170 A147524 A113177

Adjacent sequences: A241817 A241818 A241819 * A241821 A241822 A241823

Keyword

nonn,easy

Author

Clark Kimberling, Apr 30 2014

Status

approved