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%I #29 Jun 19 2023 22:35:31
%S 1,0,0,0,0,1,2,4,8,14,22,35,53,78,113,160,222,306,416,558,743,980,
%T 1281,1665,2149,2755,3514,4458,5626,7070,8846,11020,13680,16920,20852,
%U 25618,31375,38309,46649,56651,68616,82908,99940,120192,144238,172730,206425
%N Number of partitions of 2n into distinct parts < n.
%C From _Gus Wiseman_, Jun 17 2023: (Start)
%C 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:
%C () . . . . (11111) (3111) (3211) (3311) (3411)
%C (11211) (11311) (4121) (4221)
%C (12121) (11411) (5112)
%C (21112) (12221) (11511)
%C (13112) (12321)
%C (21131) (13131)
%C (21212) (13212)
%C (111122) (21231)
%C (21312)
%C (22122)
%C (31113)
%C (111141)
%C (111222)
%C (112113)
%C For partitions we have A363527, ranks A363531. For reversed partitions we have A363526, ranks A363530.
%C (End)
%e a(5) = #{4+3+2+1} = 1;
%e a(6) = #{5+4+3, 5+4+2+1} = 2;
%e a(7) = #{6+5+3, 6+5+2+1, 6+4+3+1, 5+4+3+2} = 4;
%e 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;
%e 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.
%t Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], Total[Accumulate[#]]==3n&]],{n,0,15}] (* _Gus Wiseman_, Jun 17 2023 *)
%o (Haskell)
%o a231429 n = p [1..n-1] (2*n) where
%o p _ 0 = 1
%o p [] _ = 0
%o p (k:ks) m = if m < k then 0 else p ks (m - k) + p ks m
%Y Cf. A209815, A079122.
%Y A000041 counts integer partitions, strict A000009.
%Y A053632 counts compositions by weighted sum.
%Y A264034 counts partitions by weighted sum, reverse A358194.
%Y A304818 gives weighted sum of prime indices, reverse A318283.
%Y A320387 counts multisets by weighted sum, zero-based A359678.
%Y Cf. A008284, A029931, A067538, A222855, A222955, A222970, A359042, A360672, A360675, A362559.
%K nonn
%O 0,7
%A _Reinhard Zumkeller_, Nov 14 2013
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