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A344610
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Triangle read by rows where T(n,k) is the number of integer partitions of 2n with reverse-alternating sum 2k.
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51
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1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 5, 5, 3, 1, 1, 7, 9, 6, 3, 1, 1, 11, 14, 12, 6, 3, 1, 1, 15, 23, 20, 12, 6, 3, 1, 1, 22, 34, 35, 21, 12, 6, 3, 1, 1, 30, 52, 56, 38, 21, 12, 6, 3, 1, 1, 42, 75, 91, 62, 38, 21, 12, 6, 3, 1, 1, 56, 109, 140, 103, 63, 38, 21, 12, 6, 3, 1, 1
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OFFSET
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0,4
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COMMENTS
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The reverse-alternating sum of a partition (y_1,...,y_k) is Sum_i (-1)^(k-i) y_i. This is equal to (-1)^(k-1) times the number of odd parts in the conjugate partition, where k is the number of parts.
Also the number of reversed integer partitions of 2n with alternating sum 2k.
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LINKS
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EXAMPLE
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Triangle begins:
1
1 1
2 1 1
3 3 1 1
5 5 3 1 1
7 9 6 3 1 1
11 14 12 6 3 1 1
15 23 20 12 6 3 1 1
22 34 35 21 12 6 3 1 1
30 52 56 38 21 12 6 3 1 1
42 75 91 62 38 21 12 6 3 1 1
56 109 140 103 63 38 21 12 6 3 1 1
77 153 215 163 106 63 38 21 12 6 3 1 1
Row n = 5 counts the following partitions:
(55) (442) (433) (622) (811) (10)
(3322) (541) (532) (721)
(4411) (22222) (631) (61111)
(222211) (32221) (42211)
(331111) (33211) (52111)
(22111111) (43111) (4111111)
(1111111111) (2221111)
(3211111)
(211111111)
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MATHEMATICA
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sats[y_]:=Sum[(-1)^(i-Length[y])*y[[i]], {i, Length[y]}];
Table[Length[Select[IntegerPartitions[n], k==sats[#]&]], {n, 0, 15, 2}, {k, 0, n, 2}]
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CROSSREFS
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The columns with initial 0's removed appear to converge to A006330.
The non-reversed version is A239830.
A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.
A103919 counts partitions by sum and alternating sum.
A120452 counts partitions of 2n with rev-alt sum 2 (negative: A344741).
A316524 is the alternating sum of the prime indices of n (reverse: A344616).
A344604 counts wiggly compositions with twins.
A344618 gives reverse-alternating sums of standard compositions.
Cf. A000070, A000097, A001250, A003242, A027187, A028260, A124754, A152146, A344608, A344651, A344654.
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KEYWORD
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AUTHOR
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STATUS
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approved
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