A241818 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, 4, 5, 7, 9, 12, 17, 20, 27, 37, 46, 59, 81, 102, 130, 170, 212, 273, 344, 432, 539, 679, 843, 1049, 1297, 1602, 1968, 2422, 2961, 3608, 4395, 5334, 6467, 7800, 9418, 11311, 13593, 16287, 19482, 23214, 27702, 32908, 39117, 46305, 54856, 64749

Offset

0,3

Comments

For the partition [n] of n, "max(x(i) - x(i-1))" is (as in the Mathematica program) interpreted as 0.

Links

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

Formula

a(n) + A241820(n) + A241821(n) = A000041(n) for n >= 0.

Example

a(6) counts these 7 partitions:  6, 33, 321, 222, 2211, 21111, 111111.

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. A241819, A241820, A241821, A241822, A000041.

Sequence in context: A112639 A290137 A336351 * A214120 A240840 A117599

Adjacent sequences: A241815 A241816 A241817 * A241819 A241820 A241821

Keyword

nonn,easy

Author

Clark Kimberling, Apr 30 2014

Status

approved