cp's OEIS Frontend

A344611 Number of integer partitions of 2n with reverse-alternating sum >= 0.

%I A344611 #16 Jun 12 2021 06:05:20
%S A344611 1,2,4,8,15,27,48,81,135,220,352,553,859,1313,1986,2969,4394,6439,
%T A344611 9357,13479,19273,27353,38558,53998,75168,104022,143172,196021,267051,
%U A344611 362086,488733,656802,879026,1171747,1555997,2058663,2714133,3566122,4670256,6096924,7935184
%N A344611 Number of integer partitions of 2n with reverse-alternating sum >= 0.
%C A344611 The reverse-alternating sum of a partition (y_1,...,y_k) is Sum_i (-1)^(k-i) y_i.
%C A344611 Also the number of reversed integer partitions of 2n with alternating sum >= 0.
%C A344611 The reverse-alternating sum of a partition is equal to (-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 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.
%F A344611 Conjecture: a(n) <= A160786(n). The difference is 0, 0, 0, 0, 1, 2, 4, 9, 16, 28, 48, 79, ...
%e A344611 The a(0) = 1 through a(4) = 15 partitions:
%e A344611   ()  (2)   (4)     (6)       (8)
%e A344611       (11)  (22)    (33)      (44)
%e A344611             (211)   (222)     (332)
%e A344611             (1111)  (321)     (422)
%e A344611                     (411)     (431)
%e A344611                     (2211)    (521)
%e A344611                     (21111)   (611)
%e A344611                     (111111)  (2222)
%e A344611                               (3311)
%e A344611                               (22211)
%e A344611                               (32111)
%e A344611                               (41111)
%e A344611                               (221111)
%e A344611                               (2111111)
%e A344611                               (11111111)
%t A344611 sats[y_]:=Sum[(-1)^(i-Length[y])*y[[i]],{i,Length[y]}];
%t A344611 Table[Length[Select[IntegerPartitions[n],sats[#]>=0&]],{n,0,30,2}]
%Y A344611 The non-reversed version is A058696 (partitions of 2n).
%Y A344611 The ordered version appears to be A114121.
%Y A344611 Odd bisection of A344607.
%Y A344611 Row sums of A344610.
%Y A344611 The strict case is A344650.
%Y A344611 A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.
%Y A344611 A000070 counts partitions with alternating sum 1.
%Y A344611 A000097 counts partitions with alternating sum 2.
%Y A344611 A103919 counts partitions by sum and alternating sum.
%Y A344611 A120452 counts partitions of 2n with reverse-alternating sum 2.
%Y A344611 A316524 is the alternating sum of the prime indices of n (reverse: A344616).
%Y A344611 A325534/A325535 count separable/inseparable partitions.
%Y A344611 A344612 counts partitions by sum and rev-alt sum (strict: A344739).
%Y A344611 A344618 gives reverse-alternating sums of standard compositions.
%Y A344611 A344741 counts partitions of 2n with reverse-alternating sum -2.
%Y A344611 Cf. A001250, A027187, A028260, A116406, A119899, A124754, A152146, A239829, A344608, A344609, A344649, A344651, A344654.
%K A344611 nonn
%O A344611 0,2
%A A344611 _Gus Wiseman_, May 30 2021
%E A344611 More terms from _Bert Dobbelaere_, Jun 12 2021