cp's OEIS Frontend

A357709 Number of integer partitions of n whose length is twice their alternating sum.

%I A357709 #5 Oct 17 2022 07:07:22
%S A357709 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 A357709 141,170,204,250,295,361,425,516,606,734,858,1031,1210,1440,1690,2000,
%U A357709 2347,2759,3240,3786,4441,5174,6053,7030,8210,9509,11074,12807,14870
%N A357709 Number of integer partitions of n whose length is twice their alternating sum.
%C A357709 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 A357709 The a(1) = 0 through a(12) = 6 partitions:
%e A357709   .  .  21  .  32  3111  43  3221  54      3331  65      4332
%e A357709                              4211  411111  4222  422111  4431
%e A357709                                            4321  521111  5322
%e A357709                                            5311          5421
%e A357709                                                          6411
%e A357709                                                          51111111
%t A357709 ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];
%t A357709 Table[Length[Select[IntegerPartitions[n],Length[#]==2ats[#]&]],{n,0,30}]
%Y A357709 This is the "twice" version of A357189, ranked by A357486.
%Y A357709 The version for compositions is A357847.
%Y A357709 These partitions are ranked by A357848.
%Y A357709 A000041 counts partitions, strict A000009.
%Y A357709 A025047 counts alternating compositions.
%Y A357709 A103919 counts partitions by alternating sum, full triangle A344651.
%Y A357709 A357136 counts compositions by alternating sum, full triangle A097805.
%Y A357709 A357182 counts compositions w/ length = alternating sum, ranked by A357184.
%Y A357709 Cf. A004526, A262046, A262977, A301987, A357183, A357485, A357488.
%K A357709 nonn
%O A357709 0,9
%A A357709 _Gus Wiseman_, Oct 16 2022