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

Author

Reinhard Zumkeller, Nov 14 2013

Keywords

• nonn

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)

Examples

			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.
		

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.

Programs

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

Mathematica

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