%I #6 Jun 15 2023 20:06:40

%S 1,0,0,1,0,0,2,2,0,3,3,3,5,5,10,12,7,14,25,18,22,48,48,41,67,82,89,

%T 111,140,170,220,214,264,392,386,436,623,693,756,934,1102,1301,1565,

%U 1697,2132,2616,2727,3192,4062,4550,5000,6132,7197,8067,9338,10750,12683

%N Number of integer partitions 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.

%e The a(11) = 3 through a(15) = 12 partitions (A = 10):

%e (33221) (84) (751) (662) (A5)

%e (44111) (6222) (5332) (4442) (8322)

%e (222221) (7311) (6421) (5531) (9411)

%e (621111) (532111) (43331) (722211)

%e (51111111) (42211111) (54221) (831111)

%e (65111) (3322221)

%e (432221) (3333111)

%e (443111) (4422111)

%e (32222111) (5511111)

%e (33311111) (22222221)

%e (72111111)

%e (6111111111)

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

%t Table[Length[Select[IntegerPartitions[n],altwtsum[#]==0&]],{n,0,30}]

%Y The unweighted version is A035363.

%Y These partitions have ranks A363621.

%Y The triangle for this rank statistic is A363623, reverse A363622.

%Y The version for compositions is A363626.

%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.

%K nonn

%O 0,7

%A _Gus Wiseman_, Jun 14 2023