A240451 - OEIS

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A240451

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

0, 1, 1, 2, 2, 4, 5, 8, 10, 17, 20, 28, 35, 55, 66, 87, 106, 153, 183, 243, 293, 401, 477, 608, 723, 984, 1162, 1458, 1720, 2245, 2624, 3301, 3864, 4963, 5775, 7108, 8246, 10508, 12153, 14834, 17125, 21442, 24651, 30028, 34477, 42599, 48778, 58742, 67091

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OFFSET

0,4

LINKS

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

FORMULA

a(n) + A240449(n) = A000041(n) for n >= 0.

EXAMPLE

a(6) counts these 5 partitions: 6, 51, 42, 411, 321.

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, A240449, A240452, A000041.

Sequence in context: A027193 A126796 A325831 * A206138 A241545 A157162

Adjacent sequences: A240448 A240449 A240450 * A240452 A240453 A240454

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Apr 05 2014

STATUS

approved