cp's OEIS Frontend

A357189 Number of integer partitions of n with the same length as alternating sum.

%I A357189 #11 Oct 01 2022 19:23:09
%S A357189 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 A357189 170,182,252,260,358,379,517,535,725,764,1030,1064,1427,1494,1992,
%U A357189 2059,2733,2848,3759,3887,5106,5311,6946,7177,9345,9701,12577,12996,16788
%N A357189 Number of integer partitions of n with the same length as alternating sum.
%C A357189 A partition of n is a weakly decreasing sequence of positive integers summing to n.
%C A357189 The alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i.
%e A357189 The a(4) = 1 through a(13) = 9 partitions:
%e A357189   31   311   42   322   53     333     64     443     75       553
%e A357189                   421   5111   432     5221   542     5331     652
%e A357189                                531     6211   641     6222     751
%e A357189                                51111          52211   6321     52222
%e A357189                                               62111   7311     53311
%e A357189                                                       711111   62221
%e A357189                                                                63211
%e A357189                                                                73111
%e A357189                                                                7111111
%t A357189 ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];
%t A357189 Table[Length[Select[IntegerPartitions[n],Length[#]==ats[#]&]],{n,0,30}]
%Y A357189 For product equal to sum we have A001055, compositions A335405.
%Y A357189 For product instead of length we have A004526, compositions A114220.
%Y A357189 The version for compositions is A357182, ranked by A357184.
%Y A357189 For sum equal to twice alternating sum we have A357189 (this sequence).
%Y A357189 These partitions are ranked by A357486.
%Y A357189 The reverse version is A357487, ranked by A357485.
%Y A357189 A000041 counts partitions, strict A000009.
%Y A357189 A025047 counts alternating compositions.
%Y A357189 A103919 counts partitions by alternating sum, full triangle A344651.
%Y A357189 A357136 counts compositions by alternating sum, full triangle A097805.
%Y A357189 Cf. A051159, A070939, A131044, A262046, A262977, A301987, A357183.
%K A357189 nonn
%O A357189 0,8
%A A357189 _Gus Wiseman_, Sep 30 2022