

A344611


Number of integer partitions of 2n with reversealternating sum >= 0.


48



1, 2, 4, 8, 15, 27, 48, 81, 135, 220, 352, 553, 859, 1313, 1986, 2969, 4394, 6439, 9357, 13479, 19273, 27353, 38558, 53998, 75168, 104022, 143172, 196021, 267051, 362086, 488733, 656802, 879026, 1171747, 1555997, 2058663, 2714133, 3566122, 4670256, 6096924, 7935184
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OFFSET

0,2


COMMENTS

The reversealternating sum of a partition (y_1,...,y_k) is Sum_i (1)^(ki) y_i.
Also the number of reversed integer partitions of 2n with alternating sum >= 0.
The reversealternating sum of a partition is equal to (1)^(k1) times the number of odd parts in the conjugate partition, where k is the number of parts. So a(n) is the number of partitions of 2n whose conjugate parts are all even or whose length is odd. By conjugation, this is also the number of partitions of 2n whose parts are all even or whose greatest part is odd.


LINKS



FORMULA

Conjecture: a(n) <= A160786(n). The difference is 0, 0, 0, 0, 1, 2, 4, 9, 16, 28, 48, 79, ...


EXAMPLE

The a(0) = 1 through a(4) = 15 partitions:
() (2) (4) (6) (8)
(11) (22) (33) (44)
(211) (222) (332)
(1111) (321) (422)
(411) (431)
(2211) (521)
(21111) (611)
(111111) (2222)
(3311)
(22211)
(32111)
(41111)
(221111)
(2111111)
(11111111)


MATHEMATICA

sats[y_]:=Sum[(1)^(iLength[y])*y[[i]], {i, Length[y]}];
Table[Length[Select[IntegerPartitions[n], sats[#]>=0&]], {n, 0, 30, 2}]


CROSSREFS

The nonreversed version is A058696 (partitions of 2n).
The ordered version appears to be A114121.
A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.
A000070 counts partitions with alternating sum 1.
A000097 counts partitions with alternating sum 2.
A103919 counts partitions by sum and alternating sum.
A120452 counts partitions of 2n with reversealternating sum 2.
A316524 is the alternating sum of the prime indices of n (reverse: A344616).
A344618 gives reversealternating sums of standard compositions.
A344741 counts partitions of 2n with reversealternating sum 2.
Cf. A001250, A027187, A028260, A116406, A119899, A124754, A152146, A239829, A344608, A344609, A344649, A344651, A344654.


KEYWORD

nonn


AUTHOR



EXTENSIONS



STATUS

approved



