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%I #5 Oct 17 2022 07:07:22
%S 1,0,0,1,0,1,1,1,2,2,4,3,6,6,9,11,13,18,21,28,32,44,49,65,76,96,114,
%T 141,170,204,250,295,361,425,516,606,734,858,1031,1210,1440,1690,2000,
%U 2347,2759,3240,3786,4441,5174,6053,7030,8210,9509,11074,12807,14870
%N Number of integer partitions of n whose length is twice their alternating sum.
%C The alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i. The alternating sum of a partition is also the number of odd conjugate parts.
%e The a(1) = 0 through a(12) = 6 partitions:
%e . . 21 . 32 3111 43 3221 54 3331 65 4332
%e 4211 411111 4222 422111 4431
%e 4321 521111 5322
%e 5311 5421
%e 6411
%e 51111111
%t ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];
%t Table[Length[Select[IntegerPartitions[n],Length[#]==2ats[#]&]],{n,0,30}]
%Y This is the "twice" version of A357189, ranked by A357486.
%Y The version for compositions is A357847.
%Y These partitions are ranked by A357848.
%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 A357182 counts compositions w/ length = alternating sum, ranked by A357184.
%Y Cf. A004526, A262046, A262977, A301987, A357183, A357485, A357488.
%K nonn
%O 0,9
%A _Gus Wiseman_, Oct 16 2022