A241821 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, 2, 4, 6, 10, 13, 22, 29, 40, 55, 76, 95, 129, 167, 215, 278, 354, 448, 570, 716, 896, 1115, 1387, 1713, 2116, 2597, 3182, 3881, 4741, 5748, 6976, 8416, 10177, 12219, 14704, 17592, 21051, 25101, 29960, 35559, 42267, 50017, 59253, 69898, 82524

Offset

0,6

Links

Table of n, a(n) for n=0..48.

Formula

a(n) = A241820(n) + A241822(n) for n >= 0.

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

Example

a(6) counts these 4 partitions:  51, 42, 411, 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, A241820, A241822, A000041.

Sequence in context: A258847 A238627 A336415 * A280862 A370654 A213475

Adjacent sequences: A241818 A241819 A241820 * A241822 A241823 A241824

Keyword

nonn,easy

Author

Clark Kimberling, Apr 30 2014

Status

approved