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A357639
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Number of reversed integer partitions of 2n whose half-alternating sum is 0.
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29
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1, 0, 2, 1, 6, 4, 15, 13, 37, 37, 86, 94, 194, 223, 416, 497, 867, 1056, 1746, 2159, 3424, 4272, 6546, 8215, 12248, 15418, 22449, 28311, 40415, 50985, 71543, 90222, 124730, 157132, 214392, 269696, 363733, 456739, 609611, 763969, 1010203, 1263248, 1656335, 2066552, 2688866
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OFFSET
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0,3
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COMMENTS
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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 - ...
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LINKS
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EXAMPLE
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The a(0) = 1 through a(6) = 15 reversed partitions:
() . (112) (123) (134) (145) (156)
(1111) (224) (235) (246)
(2222) (11233) (336)
(11222) (1111123) (3333)
(1111112) (11244)
(11111111) (11334)
(12333)
(1111134)
(1111224)
(1112223)
(1122222)
(11112222)
(111111222)
(11111111112)
(111111111111)
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MATHEMATICA
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halfats[f_]:=Sum[f[[i]]*(-1)^(1+Ceiling[i/2]), {i, Length[f]}];
Table[Length[Select[IntegerPartitions[2n], halfats[Reverse[#]]==0&]], {n, 0, 15}]
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CROSSREFS
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The skew-alternating version is A357640.
This is the central column of A357704.
A000041 counts integer partitions (also reversed integer partitions).
A344651 counts alternating sum of partitions by length, ordered A097805.
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.
Cf. A029862, A053251, A357189, A357487, A357488, A357631, A357634, A357636, A357639, A357641, A357645.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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