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A357704 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
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, 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, 5, 4, 8, 6, 11, 9, 3, 4, 0, 6, 0, 0, 6, 4, 11, 5, 15, 8, 13, 3, 5, 0, 7 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,6
COMMENTS
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 - ...
LINKS
EXAMPLE
Triangle begins:
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 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 5 4 8 6 11 9 3 4 0 6
0 0 6 4 11 5 15 8 13 3 5 0 7
0 0 6 5 11 8 13 19 10 13 4 5 0 7
0 0 7 5 14 8 19 13 25 9 17 4 6 0 8
0 0 7 6 14 11 19 17 29 23 13 18 5 6 0 8
Row n = 7 counts the following reversed partitions:
. . (115) (124) (133) (11113) . (7)
(1114) (1222) (223) (111112) (16)
(1123) (11122) (25)
(1111111) (34)
MATHEMATICA
halfats[f_]:=Sum[f[[i]]*(-1)^(1+Ceiling[i/2]), {i, Length[f]}];
Table[Length[Select[Reverse/@IntegerPartitions[n], halfats[#]==k&]], {n, 0, 15}, {k, -n, n, 2}]
CROSSREFS
Row sums are A000041.
Last entry of row n is A008619(n).
The central column in the non-reverse case is A035363, skew A035544.
For original reverse-alternating sum we have A344612.
For original alternating sum we have A344651, ordered A097805.
The non-reverse version is A357637, skew A357638.
The central column is A357639, skew A357640.
The non-reverse ordered version (compositions) is A357645, skew A357646.
The skew-alternating version is A357705.
A351005 = alternately equal and unequal partitions, compositions A357643.
A351006 = alternately unequal and equal partitions, compositions A357644.
A357621 gives half-alternating sum of standard compositions, skew A357623.
A357629 gives half-alternating sum of prime indices, skew A357630.
A357633 gives half-alternating sum of Heinz partition, skew A357634.
Sequence in context: A093085 A339059 A023555 * A294203 A143380 A143377
KEYWORD
nonn,tabl
AUTHOR
Gus Wiseman, Oct 10 2022
STATUS
approved

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Last modified July 13 17:55 EDT 2024. Contains 374285 sequences. (Running on oeis4.)