A231429 Number of partitions of 2n into distinct parts < n.

1, 0, 0, 0, 0, 1, 2, 4, 8, 14, 22, 35, 53, 78, 113, 160, 222, 306, 416, 558, 743, 980, 1281, 1665, 2149, 2755, 3514, 4458, 5626, 7070, 8846, 11020, 13680, 16920, 20852, 25618, 31375, 38309, 46649, 56651, 68616, 82908, 99940, 120192, 144238, 172730, 206425

Offset

0,7

Comments

From Gus Wiseman, Jun 17 2023: (Start)

Also the number of integer compositions of n with weighted sum 3*n, where the weighted sum of a sequence (y_1,...,y_k) is Sum_{i=1..k} i * y_i. The a(0) = 1 through a(9) = 14 compositions are:

() . . . . (11111) (3111) (3211) (3311) (3411)

(11211) (11311) (4121) (4221)

(12121) (11411) (5112)

(21112) (12221) (11511)

(13112) (12321)

(21131) (13131)

(21212) (13212)

(111122) (21231)

(21312)

(22122)

(31113)

(111141)

(111222)

(112113)

For partitions we have A363527, ranks A363531. For reversed partitions we have A363526, ranks A363530.

(End)

Links

Table of n, a(n) for n=0..46.

Example

a(5) = #{4+3+2+1} = 1;
a(6) = #{5+4+3, 5+4+2+1} = 2;
a(7) = #{6+5+3, 6+5+2+1, 6+4+3+1, 5+4+3+2} = 4;
a(8) = #{7+6+3, 7+6+2+1, 7+6+3, 7+5+3+1, 7+4+3+2, 6+5+4+1, 6+5+3+2, 6+4+3+2+1} = 8;
a(9) = #{8+7+3, 8+7+2+1, 8+6+4, 8+6+3+1, 8+5+4+1, 8+5+3+2, 8+4+3+2+1, 7+6+5, 7+6+4+1, 7+6+3+2, 7+5+4+2, 7+5+3+2+1, 6+5+4+3, 6+5+4+2+1} = 14.

Mathematica

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], Total[Accumulate[#]]==3n&]], {n, 0, 15}] (* Gus Wiseman, Jun 17 2023 *)

Prog

(Haskell)
a231429 n = p [1..n-1] (2*n) where
   p _  0 = 1
   p [] _ = 0
   p (k:ks) m = if m < k then 0 else p ks (m - k) + p ks m

Crossrefs

Cf. A209815, A079122.

A000041 counts integer partitions, strict A000009.

A053632 counts compositions by weighted sum.

A264034 counts partitions by weighted sum, reverse A358194.

A304818 gives weighted sum of prime indices, reverse A318283.

A320387 counts multisets by weighted sum, zero-based A359678.

Cf. A008284, A029931, A067538, A222855, A222955, A222970, A359042, A360672, A360675, A362559.

Sequence in context: A087151 A053798 A305497 * A259392 A261968 A138526

Adjacent sequences: A231426 A231427 A231428 * A231430 A231431 A231432

Keyword

nonn

Author

Reinhard Zumkeller, Nov 14 2013

Status

approved