A240449 Number of partitions p of n such that (sum of parts with multiplicity 1) <= (sum of all other parts).

1, 0, 1, 1, 3, 3, 6, 7, 12, 13, 22, 28, 42, 46, 69, 89, 125, 144, 202, 247, 334, 391, 525, 647, 852, 974, 1274, 1552, 1998, 2320, 2980, 3541, 4485, 5180, 6535, 7775, 9731, 11129, 13862, 16351, 20213, 23141, 28523, 33233, 40698, 46535, 56780, 66012, 80182

Offset

0,5

Links

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

Formula

a(n) + A240451(n) = A000041(n) for n >= 0.

Example

a(6) counts these 5 partitions:  33, 3111, 222, 2211, 21111, 111111.

Mathematica

z = 30; p[n_] := p[n] = IntegerPartitions[n]; f[p_] := f[p] = First[Transpose[p]];
ColumnForm[t = Table[Select[p[n], 2 Total[f[Select[#, Last[#] == 1 &] /. {} -> {{0, 0}}]] &[Tally[#]] < n &], {n, 0, z}]] (* shows the partitions *)
Map[Length, t]  (* A240448 *)
ColumnForm[t = Table[Select[p[n], 2 Total[f[Select[#, Last[#] == 1 &] /. {} -> {{0, 0}}]] &[Tally[#]] <= n &], {n, 0, z}]] (* shows the partitions *)
Map[Length, t]  (* A240449 *)
ColumnForm[t = Table[Select[p[n], 2 Total[f[Select[#, Last[#] == 1 &] /. {} -> {{0, 0}}]] &[Tally[#]] == n &], {n, 0, z}]] (* shows the partitions *)
Map[Length, t]  (* A240447 with alternating 0's *)
ColumnForm[t = Table[Select[p[n], 2 Total[f[Select[#, Last[#] == 1 &] /. {} -> {{0, 0}}]] &[Tally[#]] > n &], {n, 0, z}]] (* shows the partitions *)
Map[Length, t]  (* A240451 *)
ColumnForm[t = Table[Select[p[n], 2 Total[f[Select[#, Last[#] == 1 &] /. {} -> {{0, 0}}]] &[Tally[#]] >= n &], {n, 0, z}]] (* shows the partitions *)
Map[Length, t]  (* A240452 *)
(* Peter J. C. Moses, Apr 02 2014 *)

Crossrefs

Cf. A240448, A240447, A240451, A240452, A000041.

Sequence in context: A034401 A345729 A350171 * A088571 A325834 A365924

Adjacent sequences: A240446 A240447 A240448 * A240450 A240451 A240452

Keyword

nonn,easy

Author

Clark Kimberling, Apr 05 2014

Status

approved