A241828 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 parts of p.

1, 1, 2, 3, 4, 6, 8, 12, 16, 22, 30, 41, 54, 74, 97, 128, 167, 219, 280, 363, 462, 590, 746, 944, 1182, 1485, 1848, 2299, 2843, 3515, 4318, 5305, 6482, 7914, 9623, 11688, 14139, 17093, 20588, 24769, 29713, 35602, 42537, 50769, 60439, 71865, 85265, 101039

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..47.

Formula

a(n) = A241826(n) - A241825(n).

a(n) + A241823(n) + A241825(n) = A000041(n) for n >= 0.

Example

a(6) counts these 8 partitions:  6, 33, 321, 3111, 222, 2211, 21111, 111111.

Mathematica

z = 30; f[n_] := f[n] = IntegerPartitions[n]; g[p_] := Max[-Differences[p]]
Table[Count[f[n], p_ /; g[p] < Length[p]], {n, 0, z}]  (* A241828 *)
Table[Count[f[n], p_ /; g[p] <= Length[p]], {n, 0, z}] (* A241829 *)
Table[Count[f[n], p_ /; g[p] == Length[p]], {n, 0, z}] (* A241830 *)
Table[Count[f[n], p_ /; g[p] >= Length[p]], {n, 0, z}] (* A241831 *)
Table[Count[f[n], p_ /; g[p] > Length[p]], {n, 0, z}]  (* A241832 *)

Crossrefs

Cf. A241829, A241830, A241831, A241832, A000041.

Sequence in context: A180652 A046682 A005987 * A125895 A241344 A064428

Adjacent sequences: A241825 A241826 A241827 * A241829 A241830 A241831

Keyword

nonn,easy

Author

Clark Kimberling, Apr 30 2014

Status

approved