A241832 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.

0, 0, 0, 0, 0, 1, 1, 3, 3, 6, 7, 12, 14, 22, 27, 37, 46, 63, 76, 101, 124, 160, 196, 250, 302, 382, 463, 574, 693, 855, 1026, 1255, 1503, 1823, 2178, 2626, 3123, 3749, 4447, 5305, 6274, 7458, 8790, 10405, 12231, 14422, 16909, 19871, 23229, 27217, 31742

Offset

0,8

Links

Table of n, a(n) for n=0..50.

Formula

a(n) = A241831(n) - A241830(n).

a(n) + A241828(n) + A241830(n) = A000041(n) for n >= 0.

Example

a(6) counts this single partition:  51.

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. A241828, A241829, A241830, A241831, A000041.

Sequence in context: A088571 A325834 A365924 * A027187 A056508 A050065

Adjacent sequences: A241829 A241830 A241831 * A241833 A241834 A241835

Keyword

nonn,easy

Author

Clark Kimberling, Apr 30 2014

Status

approved