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A357645 Triangle read by rows where T(n,k) is the number of integer compositions of n with half-alternating sum k, where k ranges from -n to n in steps of 2.

%I A357645 #5 Oct 12 2022 09:00:48
%S A357645 1,0,1,0,0,2,0,0,1,3,0,0,2,2,4,0,0,3,5,3,5,0,0,4,8,10,4,6,0,0,5,11,18,
%T A357645 18,5,7,0,0,6,14,28,36,30,6,8,0,0,7,17,41,63,65,47,7,9,0,0,8,20,58,
%U A357645 104,126,108,70,8,10,0,0,9,23,80,164,230,230,168,100,9,11
%N A357645 Triangle read by rows where T(n,k) is the number of integer compositions of n with half-alternating sum k, where k ranges from -n to n in steps of 2.
%C A357645 We define the half-alternating sum of a sequence (A, B, C, D, E, F, G, ...) to be A + B - C - D + E + F - G - ...
%e A357645 Triangle begins:
%e A357645    1
%e A357645    0   1
%e A357645    0   0   2
%e A357645    0   0   1   3
%e A357645    0   0   2   2   4
%e A357645    0   0   3   5   3   5
%e A357645    0   0   4   8  10   4   6
%e A357645    0   0   5  11  18  18   5   7
%e A357645    0   0   6  14  28  36  30   6   8
%e A357645    0   0   7  17  41  63  65  47   7   9
%e A357645    0   0   8  20  58 104 126 108  70   8  10
%e A357645 Row n = 6 counts the following compositions:
%e A357645   (114)   (123)    (132)     (141)  (6)
%e A357645   (1113)  (213)    (222)     (231)  (15)
%e A357645   (1122)  (1212)   (312)     (321)  (24)
%e A357645   (1131)  (1221)   (1311)    (411)  (33)
%e A357645           (2112)   (2211)           (42)
%e A357645           (2121)   (3111)           (51)
%e A357645           (11121)  (11112)
%e A357645           (11211)  (12111)
%e A357645                    (21111)
%e A357645                    (111111)
%t A357645 halfats[f_]:=Sum[f[[i]]*(-1)^(1+Ceiling[i/2]),{i,Length[f]}];
%t A357645 Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],halfats[#]==k&]],{n,0,10},{k,-n,n,2}]
%Y A357645 Row sums are A011782.
%Y A357645 For original alternating sum we have A097805, unordered A344651.
%Y A357645 Column k = n-4 appears to be A177787.
%Y A357645 The case of partitions is A357637, skew A357638.
%Y A357645 The central column k=0 is A357641 (aerated).
%Y A357645 The skew-alternating version is A357646.
%Y A357645 The reverse version for partitions is A357704, skew A357705.
%Y A357645 A351005 = alternately equal and unequal partitions, compositions A357643.
%Y A357645 A351006 = alternately unequal and equal partitions, compositions A357644.
%Y A357645 A357621 gives half-alternating sum of standard compositions, skew A357623.
%Y A357645 A357629 gives half-alternating sum of prime indices, skew A357630.
%Y A357645 A357633 gives half-alternating sum of Heinz partition, skew  A357634.
%Y A357645 Cf. A029862, A035363, A035544, A357136, A357631, A357639.
%K A357645 nonn,tabl
%O A357645 0,6
%A A357645 _Gus Wiseman_, Oct 12 2022