The OEIS is supported by the many generous donors to the OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A344607 Number of integer partitions of n with reverse-alternating sum >= 0. 61
 1, 1, 2, 2, 4, 4, 8, 8, 15, 16, 27, 29, 48, 52, 81, 90, 135, 151, 220, 248, 352, 400, 553, 632, 859, 985, 1313, 1512, 1986, 2291, 2969, 3431, 4394, 5084, 6439, 7456, 9357, 10836, 13479, 15613, 19273, 22316, 27353, 31659, 38558, 44601, 53998, 62416, 75168 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS The reverse-alternating sum of a partition (y_1,...,y_k) is Sum_i (-1)^(k-i) y_i. Also the number of reversed integer partitions of n with alternating sum >= 0. A formula for the reverse-alternating sum of a partition is: (-1)^(k-1) times the number of odd parts in the conjugate partition, where k is the number of parts. So a(n) is the number of integer partitions of n whose conjugate parts are all even or whose length is odd. By conjugation, this is also the number of integer partitions of n whose parts are all even or whose greatest part is odd. All integer partitions have alternating sum >= 0, so the non-reversed version is A000041. Is this sequence weakly increasing? In particular, is A344611(n) <= A160786(n)? LINKS FORMULA a(n) + A344608(n) = A000041(n). a(2n+1) = A160786(n). EXAMPLE The a(1) = 1 through a(8) = 15 partitions:   (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)        (11)  (111)  (22)    (221)    (33)      (322)      (44)                     (211)   (311)    (222)     (331)      (332)                     (1111)  (11111)  (321)     (421)      (422)                                      (411)     (511)      (431)                                      (2211)    (22111)    (521)                                      (21111)   (31111)    (611)                                      (111111)  (1111111)  (2222)                                                           (3311)                                                           (22211)                                                           (32111)                                                           (41111)                                                           (221111)                                                           (2111111)                                                           (11111111) MATHEMATICA sats[y_]:=Sum[(-1)^(i-Length[y])*y[[i]], {i, Length[y]}]; Table[Length[Select[IntegerPartitions[n], sats[#]>=0&]], {n, 0, 30}] CROSSREFS The non-reversed version is A000041. The opposite version (rev-alt sum <= 0) is A027187, ranked by A028260. The strict case for n > 0 is A067659 (even bisection: A344650). The ordered version appears to be A116406 (even bisection: A114121). The odd bisection is A160786. The complement is counted by A344608. The Heinz numbers of these partitions are A344609 (complement: A119899). The even bisection is A344611. A000070 counts partitions with alternating sum 1 (reversed: A000004). A000097 counts partitions with alternating sum 2 (reversed: A120452). A035363 counts partitions with alternating sum 0, ranked by A000290. A103919 counts partitions by sum and alternating sum. A316524 is the alternating sum of prime indices of n (reversed: A344616). A325534/A325535 count separable/inseparable partitions. A344610 counts partitions by sum and positive reverse-alternating sum. A344612 counts partitions by sum and reverse-alternating sum. A344618 gives reverse-alternating sums of standard compositions. Cf. A006330, A071321, A071322, A124754, A239829, A239830, A344604, A344651, A344654, A344739, A344742. Sequence in context: A262966 A034397 A200750 * A325722 A279818 A263614 Adjacent sequences:  A344604 A344605 A344606 * A344608 A344609 A344610 KEYWORD nonn AUTHOR Gus Wiseman, May 29 2021 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified January 21 10:25 EST 2022. Contains 350477 sequences. (Running on oeis4.)