A357704 - OEIS

%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