%I #11 Oct 01 2022 19:23:09

%S 1,1,0,0,1,1,1,2,2,4,3,5,6,9,9,13,16,23,23,34,37,54,54,78,84,120,121,

%T 170,182,252,260,358,379,517,535,725,764,1030,1064,1427,1494,1992,

%U 2059,2733,2848,3759,3887,5106,5311,6946,7177,9345,9701,12577,12996,16788

%N Number of integer partitions of n with the same length as alternating sum.

%C A partition of n is a weakly decreasing sequence of positive integers summing to n.

%C The alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i.

%e The a(4) = 1 through a(13) = 9 partitions:

%e 31 311 42 322 53 333 64 443 75 553

%e 421 5111 432 5221 542 5331 652

%e 531 6211 641 6222 751

%e 51111 52211 6321 52222

%e 62111 7311 53311

%e 711111 62221

%e 63211

%e 73111

%e 7111111

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

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

%Y For product equal to sum we have A001055, compositions A335405.

%Y For product instead of length we have A004526, compositions A114220.

%Y The version for compositions is A357182, ranked by A357184.

%Y For sum equal to twice alternating sum we have A357189 (this sequence).

%Y These partitions are ranked by A357486.

%Y The reverse version is A357487, ranked by A357485.

%Y A000041 counts partitions, strict A000009.

%Y A025047 counts alternating compositions.

%Y A103919 counts partitions by alternating sum, full triangle A344651.

%Y A357136 counts compositions by alternating sum, full triangle A097805.

%Y Cf. A051159, A070939, A131044, A262046, A262977, A301987, A357183.

%K nonn

%O 0,8

%A _Gus Wiseman_, Sep 30 2022