A241819 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.

1, 1, 2, 3, 5, 6, 9, 11, 17, 20, 30, 37, 50, 64, 84, 106, 141, 178, 224, 290, 368, 457, 574, 722, 894, 1113, 1371, 1693, 2082, 2555, 3108, 3806, 4630, 5605, 6787, 8197, 9881, 11877, 14256, 17047, 20395, 24320, 28958, 34409, 40867, 48333, 57243, 67548, 79683

Offset

0,3

Links

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

Formula

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

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

Example

a(6)= 9 counts all of the 11 partitions of 6 except 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, A241820, A241821, A241822, A000041.

Sequence in context: A131995 A363066 A060714 * A227070 A032718 A366143

Adjacent sequences: A241816 A241817 A241818 * A241820 A241821 A241822

Keyword

nonn,easy

Author

Clark Kimberling, Apr 30 2014

Status

approved