A241829 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, 5, 6, 10, 12, 19, 24, 35, 44, 63, 79, 108, 139, 185, 234, 309, 389, 503, 632, 806, 1005, 1273, 1576, 1973, 2436, 3025, 3710, 4578, 5587, 6846, 8320, 10132, 12257, 14854, 17888, 21568, 25880, 31064, 37125, 44384, 52856, 62944, 74712, 88649, 104883

Offset

0,3

Links

Table of n, a(n) for n=0..47.

Formula

a(n) = A241828(n) + A241830(n).

a(n) + A241832(n) = A000041(n) for n >= 0.

Example

a(6) counts these 10 partitions:  6, 42, 411, 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. A241828, A241830, A241831, A241832, A000041.

Sequence in context: A237831 A347443 A329235 * A339511 A250179 A202823

Adjacent sequences: A241826 A241827 A241828 * A241830 A241831 A241832

Keyword

nonn,easy

Author

Clark Kimberling, Apr 30 2014

Status

approved