A241826 Number of partitions p = [x(1), ..., x(k)], where x(1) >= x(2) >= ... >= x(k), of n such that min(x(i) - x(i-1)) >= number of distinct parts of p.

1, 1, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 7, 7, 9, 10, 12, 13, 16, 17, 20, 22, 25, 27, 31, 33, 37, 40, 45, 48, 54, 58, 65, 70, 78, 84, 94, 101, 112, 121, 134, 144, 159, 171, 188, 202, 221, 237, 259, 277, 301, 322, 349, 372, 402, 429, 462, 492, 529, 563, 605, 643

Offset

0,5

Comments

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

Links

Table of n, a(n) for n=0..61.

Formula

a(n) = A241825(n) + A241827(n).

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

Example

a(6) counts these 3 partitions:  6, 51, 42.

Mathematica

z = 30; f[n_] := f[n] = IntegerPartitions[n]; d[p_] := d[p] = Length[DeleteDuplicates[p]]; g1[p_] := Min[-Differences[p]]
Table[Count[f[n], p_ /; g1[p] < d[p]], {n, 0, z}]  (* A241823 *)
Table[Count[f[n], p_ /; g1[p] <= d[p]], {n, 0, z}] (* A241824 *)
Table[Count[f[n], p_ /; g1[p] == d[p]], {n, 0, z}] (* A241825 *)
Table[Count[f[n], p_ /; g1[p] >= d[p]], {n, 0, z}] (* A241826 *)
Table[Count[f[n], p_ /; g1[p] > d[p]], {n, 0, z}]  (* A241827 *)

Crossrefs

Cf. A241823, A241824, A241826, A241827, A000041.

Sequence in context: A188666 A328090 A029156 * A237979 A264591 A026826

Adjacent sequences: A241823 A241824 A241825 * A241827 A241828 A241829

Keyword

nonn,easy

Author

Clark Kimberling, Apr 30 2014

Status

approved