%I #16 Sep 05 2023 15:21:20

%S 1,0,0,1,1,0,2,5,7,8,14,38,64,87,174,373,649,1069,2051,4091,7453,

%T 13276,25260,48990,91378,168890,321661,618323,1169126,2203649,4211163,

%U 8085240,15421171,29390131,56382040,108443047,208077560,399310778

%N Number of integer compositions of n with weighted alternating sum 0.

%C We define the weighted alternating sum of a sequence (y_1,...,y_k) to be Sum_{i=1..k} (-1)^(i-1) * i * y_i.

%H Alois P. Heinz, <a href="/A363626/b363626.txt">Table of n, a(n) for n = 0..150</a> (first 51 terms from Max Alekseyev)

%e The a(3) = 1 through a(10) = 14 compositions:

%e (21) (121) . (42) (331) (242) (63) (541)

%e (3111) (1132) (1331) (153) (2143)

%e (2221) (11132) (4122) (3232)

%e (21121) (12221) (5211) (4321)

%e (112111) (23111) (13122) (15112)

%e (121121) (14211) (31231)

%e (1112111) (411111) (42121)

%e (1311111) (114112)

%e (212122)

%e (213211)

%e (311221)

%e (322111)

%e (3111121)

%e (21211111)

%t altwtsum[y_]:=Sum[(-1)^(k-1)*k*y[[k]],{k,1,Length[y]}];

%t Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],altwtsum[#]==0&]],{n,0,10}]

%Y The unweighted version is A138364, ranks A344619.

%Y The version for partitions is A363532, ranks A363621.

%Y A000041 counts integer partitions.

%Y A264034 counts partitions by weighted sum, reverse A358194.

%Y A304818 gives weighted sum of prime indices, reverse A318283.

%Y A316524 gives alternating sum of prime indices, reverse A344616.

%Y A363619 gives weighted alternating sum of prime indices, reverse A363620.

%Y A363624 gives weighted alternating sum of Heinz partition, reverse A363625.

%Y Cf. A008284, A053632, A106529, A261079, A320387, A360672, A360675, A362559, A363622, A363623.

%K nonn

%O 0,7

%A _Gus Wiseman_, Jun 16 2023

%E Terms a(22) onward from _Max Alekseyev_, Sep 05 2023