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Triangle read by rows where T(n,k) is the number of reversed integer partitions of n with half-alternating sum k, where k ranges from -n to n in steps of 2.
8

%I #7 Oct 10 2022 20:47:12

%S 1,0,1,0,0,2,0,0,1,2,0,0,2,0,3,0,0,2,2,0,3,0,0,3,1,3,0,4,0,0,3,2,4,2,

%T 0,4,0,0,4,2,6,2,3,0,5,0,0,4,3,5,7,3,3,0,5,0,0,5,3,8,4,10,2,4,0,6,0,0,

%U 5,4,8,6,11,9,3,4,0,6,0,0,6,4,11,5,15,8,13,3,5,0,7

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

%C 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 Triangle begins:

%e 1

%e 0 1

%e 0 0 2

%e 0 0 1 2

%e 0 0 2 0 3

%e 0 0 2 2 0 3

%e 0 0 3 1 3 0 4

%e 0 0 3 2 4 2 0 4

%e 0 0 4 2 6 2 3 0 5

%e 0 0 4 3 5 7 3 3 0 5

%e 0 0 5 3 8 4 10 2 4 0 6

%e 0 0 5 4 8 6 11 9 3 4 0 6

%e 0 0 6 4 11 5 15 8 13 3 5 0 7

%e 0 0 6 5 11 8 13 19 10 13 4 5 0 7

%e 0 0 7 5 14 8 19 13 25 9 17 4 6 0 8

%e 0 0 7 6 14 11 19 17 29 23 13 18 5 6 0 8

%e Row n = 7 counts the following reversed partitions:

%e . . (115) (124) (133) (11113) . (7)

%e (1114) (1222) (223) (111112) (16)

%e (1123) (11122) (25)

%e (1111111) (34)

%t halfats[f_]:=Sum[f[[i]]*(-1)^(1+Ceiling[i/2]),{i,Length[f]}];

%t Table[Length[Select[Reverse/@IntegerPartitions[n],halfats[#]==k&]],{n,0,15},{k,-n,n,2}]

%Y Row sums are A000041.

%Y Last entry of row n is A008619(n).

%Y The central column in the non-reverse case is A035363, skew A035544.

%Y For original reverse-alternating sum we have A344612.

%Y For original alternating sum we have A344651, ordered A097805.

%Y The non-reverse version is A357637, skew A357638.

%Y The central column is A357639, skew A357640.

%Y The non-reverse ordered version (compositions) is A357645, skew A357646.

%Y The skew-alternating version is A357705.

%Y A351005 = alternately equal and unequal partitions, compositions A357643.

%Y A351006 = alternately unequal and equal partitions, compositions A357644.

%Y A357621 gives half-alternating sum of standard compositions, skew A357623.

%Y A357629 gives half-alternating sum of prime indices, skew A357630.

%Y A357633 gives half-alternating sum of Heinz partition, skew A357634.

%Y Cf. A029862, A053251, A357136, A357189, A357487, A357488, A357631, A357632, A357641.

%K nonn,tabl

%O 0,6

%A _Gus Wiseman_, Oct 10 2022