cp's OEIS Frontend

A357183 Number of integer compositions with the same length as the absolute value of their alternating sum.

%I A357183 #13 Sep 29 2022 12:56:36
%S A357183 1,1,0,0,2,3,2,5,12,22,26,58,100,203,282,616,962,2045,2982,6518,9858,
%T A357183 21416,31680,69623,104158,228930,339978,751430,1119668,2478787,
%U A357183 3684082,8182469,12171900,27082870,40247978,89748642,133394708,297933185,442628598,990210110
%N A357183 Number of integer compositions with the same length as the absolute value of their alternating sum.
%C A357183 A composition of n is a finite sequence of positive integers summing to n.
%C A357183 The alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i.
%e A357183 The a(1) = 1 through a(8) = 12 compositions:
%e A357183   (1)  (13)  (113)  (24)  (124)  (35)
%e A357183        (31)  (212)  (42)  (151)  (53)
%e A357183              (311)        (223)  (1115)
%e A357183                           (322)  (1151)
%e A357183                           (421)  (1214)
%e A357183                                  (1313)
%e A357183                                  (1412)
%e A357183                                  (1511)
%e A357183                                  (2141)
%e A357183                                  (3131)
%e A357183                                  (4121)
%e A357183                                  (5111)
%t A357183 ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];
%t A357183 Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],Length[#]==Abs[ats[#]]&]],{n,0,15}]
%Y A357183 For product instead of length we have A114220.
%Y A357183 For sum equal to twice alternating sum we have A262977, ranked by A348614.
%Y A357183 For product equal to sum we have A335405, ranked by A335404.
%Y A357183 This is the absolute value version of A357182.
%Y A357183 These compositions are ranked by A357185.
%Y A357183 The case of partitions is A357189.
%Y A357183 A003242 counts anti-run compositions, ranked by A333489.
%Y A357183 A011782 counts compositions.
%Y A357183 A025047 counts alternating compositions, ranked by A345167.
%Y A357183 A124754 gives alternating sums of standard compositions.
%Y A357183 A238279 counts compositions by sum and number of maximal runs.
%Y A357183 A261983 counts non-anti-run compositions.
%Y A357183 A357136 counts compositions by alternating sum.
%Y A357183 Cf. A000120, A032020, A070939, A106356, A114901, A131044, A178470, A233564, A242882, A262046, A301987.
%K A357183 nonn
%O A357183 0,5
%A A357183 _Gus Wiseman_, Sep 28 2022
%E A357183 a(21)-a(39) from _Alois P. Heinz_, Sep 29 2022