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A357646 Triangle read by rows where T(n,k) is the number of integer compositions of n with skew-alternating sum k, where k ranges from -n to n in steps of 2.

%I A357646 #5 Oct 12 2022 19:44:46
%S A357646 1,0,1,0,1,1,0,2,1,1,0,3,3,1,1,0,4,5,5,1,1,0,5,7,10,8,1,1,0,6,9,17,18,
%T A357646 12,1,1,0,7,11,27,35,29,17,1,1,0,8,13,41,63,63,43,23,1,1,0,9,15,60,
%U A357646 106,126,104,60,30,1,1,0,10,17,85,168,232,230,162,80,38,1,1
%N A357646 Triangle read by rows where T(n,k) is the number of integer compositions of n with skew-alternating sum k, where k ranges from -n to n in steps of 2.
%C A357646 We define the skew-alternating sum of a sequence (A, B, C, D, E, F, G, ...) to be A - B - C + D + E - F - G + ...
%e A357646 Triangle begins:
%e A357646    1
%e A357646    0   1
%e A357646    0   1   1
%e A357646    0   2   1   1
%e A357646    0   3   3   1   1
%e A357646    0   4   5   5   1   1
%e A357646    0   5   7  10   8   1   1
%e A357646    0   6   9  17  18  12   1   1
%e A357646    0   7  11  27  35  29  17   1   1
%e A357646    0   8  13  41  63  63  43  23   1   1
%e A357646    0   9  15  60 106 126 104  60  30   1   1
%e A357646 Row n = 6 counts the following compositions:
%e A357646   (15)   (24)    (33)      (42)     (51)  (6)
%e A357646   (114)  (213)   (312)     (411)
%e A357646   (123)  (222)   (321)     (1113)
%e A357646   (132)  (231)   (1122)    (2112)
%e A357646   (141)  (1131)  (1212)    (3111)
%e A357646          (1221)  (2121)    (11112)
%e A357646          (1311)  (2211)    (11121)
%e A357646                  (11211)   (21111)
%e A357646                  (12111)
%e A357646                  (111111)
%t A357646 skats[f_]:=Sum[f[[i]]*(-1)^(1+Ceiling[(i+1)/2]),{i,Length[f]}];
%t A357646 Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],skats[#]==k&]],{n,0,10},{k,-n,n,2}]
%Y A357646 The central column k=0 is A001700 (aerated), half A357641.
%Y A357646 Row sums are A011782.
%Y A357646 For original alternating sum we have A097805, unordered A344651.
%Y A357646 The skew-alternating sum of standard compositions is A357623, half A357621.
%Y A357646 The case of partitions is A357638, half A357637.
%Y A357646 The half-alternating version is A357645.
%Y A357646 The reverse version for partitions is A357705, half A357704.
%Y A357646 Cf. A029862, A035363, A035544, A177787, A357136, A357630, A357631, A357634, A357639, A357643, A357644.
%K A357646 nonn,tabl
%O A357646 0,8
%A A357646 _Gus Wiseman_, Oct 12 2022